%                   from DEFS.TEX:
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%                   main file begins:
{\noindent
{\obeylines\twelvebf\baselineskip=20pt
Symmetries of Distributions and Quadrature of
Ordinary Differential Equations}
\vskip 1cm
{\baselineskip 10pt {\obeylines
S. V. Duzhin
{\eightit
Program Systems Institute of the USSR Academy of Sciences,
Pereslavl-Zalessky, 152140, USSR}
{\eighttt E-mail: psi@cernvax.cern.ch}
\medskip
and
\medskip
V. V. Lychagin
{\eightit
All-Union Correspondence Institute
of Civil Engineering, Sr. Kalitnikovskaya 30,
Moscow, 109807, USSR}}% end of \obeylines instruction
\vskip 1cm
{\ninebf Abstract.}
{\ninerm We present a geometric exposition of S.Lie's and E.Cartan's
theory of explicit integration of finite type (in particular,
ordinary) differential equations. Numerous examples of how this theory
works are given. In one of these, we propose a method of hunting for
particular solutions of partial differential equations via
symmetry preserving overdetermination.}
\medskip
{\ninebf AMS subject classification (1980).} {\ninerm 34A05,
35C05, 53C12, 58A30, 17B30.}
\medskip
{\ninebf Key words.} {\ninerm Distribution, symmetry, explicit
solution, ordinary differential equation, finite type system,
solvable Lie algebra.}
}%end of \baselineskip instruction
}%end of \noindent  instruction

\beginsection            Introduction

     It is widely known that the notion of a solvable Lie algebra
is closely associated with differential equations solvable by
quadratures and that this theory goes back to Sophus Lie.
However, S.~Lie himself never stated the general theorem
underlying this fact.
His main book on the subject [Lie] is a collection of geometrical
algorithms for solving
various classes of differential equations, unified by the idea
of symmetry group.
S.~Lie managed to explain the whole variety of existing
integration tricks from a uniform stand-point
and also to create new powerful methods of quadrature and reduction.

Symmetries of differential equations can be used to reduce the
dimension of the problem under study in either of the two ways,
by passing over to quotient manifolds (first integration
strategy---reduction of order) or by restricting the problem to a
submanifold (second integration strategy---first integrals).
One of the examples of the second integration method is provided
by Theorem 49 of [Lie], which says that
a linear first order partial differential equation in 4
variables (equivalent to a non-autonomous system of
3 o.d.e.'s) is reduced to quadratures, if it has a solvable
3-dimensional symmetry group satisfying certain non-degeneracy
conditions. Propositions of this kind,
apparently general in nature, do not appear in [Lie]
as a universal theorem.
As writes E.~M.~Polishchuk in his biographical book [Pol],
`this theorem seems to be dissolved in the totality of other
Lie's results about groups admitted by equations'.

The second integration method of Lie was further developed
by Elie Cartan who thought about the symmetries of
arbitrary Pfaff systems (=distributions) in terms of vector fields and
differential forms.
Although E.~Cartan, no doubt, understood the general fact
equivalent to Theorem~3 below, he never stated it in [Ca].
Our aim is to clarify the results of Lie and Cartan placing them
in the general differential-geometric context
of manifolds and distributions [KN, St].  We give a precise
statement of this theorem in a form that equally
applies to single ordinary differential equations, systems of
o.d.e.'s and to arbitrary systems of partial differential
equations of finite type.

During the last 30 years, beginning with the pioneering book by
L.~V.~Ovsiannikov [Ovs] whose Russian edition was published in
1962, an extensive literature on the symmetry groups of
differential equations has appeared (cf. [Ib], [KLM], [Ol], [BK],
[Sph]).  We especially recommend the book by H.~Stephani [Sph]
which contains precise recipes of integration procedures,
numerous suggestive examples, informal motivations and is very
close to the original S.~Lie's papers both in contents and style.
In particular, it is as informal as possible to be read by people
anxious just to solve their favourite ordinary differential
equations.

Although many authors have nicely explained and developed S.~Lie's
approach, especially as regards his first integration method,
there does not seem to exist any modern exposition of
the relevant work of Elie Cartan who put the theorem of Lie into
a more general setting. The books listed above do not unveil
the differential-geometric background that unifies the second
integration method as applied to
ordinary differential equations, systems of such and finite type
systems of partial differential equations.
It goes without saying that specialists in group
applications to differential equations know the Lie
and Cartan's theorem in full generality, yet nobody will give you
an exact reference to a written mathematical text that contains its
precise statement.
Our primary aim is to
fill this gap.\footnote*{\eightrm While revising the manuscript,
the authors came across the research report by J.~Sherring and
G.~Prince [SP] where Cartan's ideas are discussed in a spirit
close to our own, but with applications only to systems of ordinary
differential equations.}

In sections 2 to 6, our style follows that of [BK], [Sph]
(and also the classical literature of the beginning of the
century) in that we do not state the exact notion of {\sl
integration by quadratures\/}, although it is crucial in the
formulation of the main theorem.  By doing so, we just adhere to
the state of the art in this area.
For the case of {\sl linear algebraic\/}
differential equations, the notion of quadratures is readily
formalized in the framework of differential algebra (see [Ko],
[Po2]). But, as far as the authors know, until now there have not
been any successful formulations of nonlinear differential
algebra that could meet our purposes. See [Cas] for one of the
recent attempts to clarify the interrelation between differential
geometry and differential algebra.

The paper proceeds as follows.
      In section 1, we recall basic definitions pertaining to
distributions on manifolds and introduce the notion of symmetry.
In section~2, we explain how to find first integrals of
distributions using their symmetries. This theory is specialized
for ordinary differential equations and finite type differential
systems in sections 3 and 4, respectively.
Section~5 contains some illustrative examples of integration of
ordinary differential equations by means of the developed theory.
Finally, in Section~6 we propose a trick which allows to find
finite-parametric families of solutions of partial differential
equations with ample symmetry algebras by reducing the problem to
that of integration of a completely integrable distribution.

\beginsection 1. Symmetries of distributions

    We begin by recalling some basic facts about distributions (see [KN],
[St] for details).

     Let~$P$ be a distribution of dimension~$m$ and codimension~$k$ on a
smooth manifold~$M$. This means that ${\dim M = m+k}$ and at every
point~$x$ of~$M$ an $m$-dimensional subspace $P_x$ of the tangent space
$T_x M$ is specified in such a way thet $P_x$ smoothly depends on $x$.  Let
  $$D(P) = \{\,V \in \Vect(M) \mid
          V_x \in P_x \ \forall x \in M\}$$
be the set of all smooth vector fields lying in the distribution $P$ and
  $$\Lambda(P) = \{\,\omega \in \Omega^1(M) \mid
          \omega(V)=0 \ \forall V  \in D(P) \,\}$$
the set of all differential 1-forms vanishing on vectors from~$P$.

    Both $D(P)$ and $\Lambda(P)$ are modules over the ring of
smooth functions $C^\infty(M)$ and a distribution is fixed whenever
one specifies a system of generators of either of these modules.
Both modules are usually free (this is always the case with their
localizations to an appropriate open everywhere dense subset of $M$), so
all one has to specify is either a set $V_1,\ldots,V_m$ of vector fields
whose values at every point of the manifold are linearly independent or
a set $\omega_1,\ldots,\omega_k$ of 1-forms subject to the same
requirement.
\medskip
   {\bf Example 1.} The 1-form $\omega=dy-zdx$ does not vanish in any
point of the 3-space ${\bf R}^3$, so it defines a distribution of dimension
2 and codimension 1. For reasons explained below, it is called
{\sl Cartan's distribution}. A basis of the module $D(P)$ is formed by vector
fields $V_1=\der_x+z\der_y$ and $V_2=\der_z$.

   {\bf Example 2.} Let $M={\bf R}\times{\bf R}^+\times S^1$ with
coordinates $x\in{\bf R}$, $y\in{\bf R}^+$, $\varphi\in S^1={\bf R} \bmod 2\pi$.
The 1-form
$$\omega=2\sin^2\negthinspace{\varphi\over 2}\thinspace dx
             +\sin\varphi\thinspace dy-y\thinspace d\varphi$$
defines the so called {\sl oricycle distribution} (explanation below).
In this example both modules $D(P)$ and $\Lambda(P)$ are again free.

   {\bf Example 3.} We will describe this one in plain English, without
formulas. Imagine a M\"obius band and at every point of its surface the line,
lying in the tangent plane and
perpendicular to the central circle of the band. Both modules
$D(P)$ and $\Lambda(P)$
associated with this distribution are not free, because the fibering of the
M\"obius band over its central circle is non-trivial. But both modules become
free as soon as we cut the band in such a way that it becomes
topologically trivial.
\medskip

A distribution is by definition an infinitesimal object. It is connected with
the world of finite things via the notion of {\sl integral manifold}.
 A submanifold $N\subset M$ is said to be
integral for the distribution $P$ if $T_x N \subset P_x$ for all
$x \in M$. An integral manifold is {\sl maximal} if it is not contained in an
integral manifold of greater dimension. For example, lines on the surface
of the M\"obius band which are perpendicular to the base circle, are
maximal integral manifolds for the above mentioned distribution.

A distribution is said to be {\sl completely integrable} if the dimension
of every maximal integral manifold is exactly $m$, the dimension of the
distribution itself. In this case the entire manifold is the disjoint
union of maximal integral manifolds of the distribution, which are the
{\sl leaves} of a {\sl foliation}, so that the notion
of a completely integrable distribution is equivalent to that of a
 foliation.

The striped M\"obius band of Example 3 above is obviously a foliation
(see Fig.~1).
It is somewhat more difficult to decide whether the distributions of Examples
1 and 2 are completely integrable. The classical Frobenius' theorem gives
two equivalent criteria for checking the complete integrability
of a distribution. They are: 1)~$D(P)$ should be a Lie algebra, and 2)~the
differential of any 1-form from $\Lambda(P)$ should belong to the ideal
generated by $\Lambda(P)$. In the case of a distribution
defined by a single 1-form $\omega$ the second condition simply means that
$\omega\wedge d\omega=0$.

The reader can readily verify that this condition holds for Example 2 and
fails for Example 1. Maximal integral manifolds for the Cartan distribution
are 1-dimensional; they will be described later in section 3.
Maximal integral manifolds of the oricycle distribution are 2-dimensional;
they can be easily found in a straightforward manner, but we will do that
in a more elegant way --- using symmetries --- just to illustrate the meaning
and the scope of applications of the latter.

By a (finite) {\sl symmetry} of the distribution $P$ we understand a
(possibly local)
 diffeomorphism $f:M\rightarrow M$ which takes $P$ into itself, i.e.
such that  $f_*(P_x) \subset P_{f(x)}$ for all $x\in M$.
A vector field~$X$ is said to be an (infinitesimal) symmetry of the
distribution if the flow generated by~$X$ consists of finite symmetries.
The infinitesimal approach turns out to be much more constructive than
its finite counterpart, so in what follows
the word {\sl symmetry} will always mean {\sl infinitesimal symmetry} unless
otherwise explicitely specified.
\medskip
    {\bf Example 4.}
In Examples 1 and 2 given above, the vector field $\der_x$ is
a symmetry, because in both examples
the coefficients of the basic 1-form $\omega$ do not depend
on $x$ and hence the corresponding finite transformations, i.e.
translations in $x$, preserve these forms. In example 3, the
infinitesimal rotation of the M\"obius band along its center circle
is also a symmetry.
\medskip
    The set of all symmetries $\Sym(P)$ forms
a Lie algebra with respect to vector fields commutator, because finite
symmetries obviously make a group. The following theorem allows one to
deduce this fact by a simple manipulation with formulas. It gives two
constructive characterizations of the symmetry algebra, one in terms of
the associated module of vector fields $D(P)$, and another in terms of the
dual module of 1-forms $\Lambda(P)$.
\medskip
{\bf Theorem 1.} {\sl The following conditions are equivalent\/}:
\smallskip
(1)\ \ $X \in \Sym (P)$,

(2)\ \ $[X,D(P)] \subset D(P)$,

(3)\ \ $L_X(\Lambda(P)) \subset \Lambda(P)$,
\smallskip
{\sl where $L_X$ is the Lie derivative operator along the vector field $X$.}
\medskip

\bf Proof. \rm Choose some (possibly local) bases $V_1,\ldots,V_m$ and
$\omega_1,\ldots,\omega_k$ in the $C^\infty (M)$-modules $D(P)$ and
$\Lambda(P)$, respectively. Conditions (2) and (3) can be rewritten as
follows:

(2) there exist such functions $\alpha_{ij}$ that for all $i$ one has
    $$[X,V_i]=\sum_{j=1}^m \alpha_{ij}V_j,$$

(3) there exist such functions $\beta_{ij}$ that for all $i$ one has
    $$L_X(\omega_i)=\sum_{j=1}^k \beta_{ij}\omega_j,$$

The theorem follows from the three implications
$(1)\Rightarrow(2)$, $(2)\Rightarrow(3)$, $(3)\Rightarrow(1)$, and we are
going to prove them succesively.

$(1)\Rightarrow(2)$. Let $\{G_t\}$ be the 1-parametric transformation group
corresponding to the vector field~$X$ which is a symmetry of~$P$.
Then for every time moment
$t\in \bf R$ and all $i=1,\ldots,m$ we have
    $$(G_t)_*(V_i)=\sum_{j=1}^m \lambda_{ij}(t)V_j,$$
where $\lambda_{ij}$ is a family of smooth functions on the manifold $M$
smoothly depending on the parameter $t$.

Differentiating this over $t$ at $t=0$ and using the formula
    $$L_X(V)=-{d\over dt}\Big|_{t=0}(G_t)_*(V),$$
one gets
    $$[X,V_i]=\sum_{j=1}^m \alpha_{ij}V_j,$$
with
    $$\alpha_{ij}=-{d\over dt}\Big|_{t=0}\lambda_{ij}(t).$$

$(2)\Rightarrow(3)$. Suppose that a vector field $X$ satisfies (2).
Now take a 1-form $\omega$ which vanishes on all the vector fields
$V_1,\ldots,V_m$ and prove that $L_X(\omega)$ has the same property
(and hence can be represented as a linear combination of
$\omega_1,\ldots,\omega_k$). Indeed, for all $i=1,\ldots,m$ we have

$$\eqalign{L_X\bigl(\omega(V_i)\bigr)
              &= \bigl(X\cchrg d\omega+d(X\cchrg d\omega)\bigr)(V_i)\cr
              &= d\omega(X,V_i)+V_i(\omega(X)) \cr
              &= X\bigl(\omega(V_i)\bigr)-\omega\bigl([X,V_i]\bigr)=0\cr}$$

$(3)\Rightarrow(1)$. Consider the following differential $(k+1)$-forms,
dependent upon the parameter $t$:
$$\Omega_i(t)=G_t^*(\omega_i)\wedge\omega_1\wedge\ldots\wedge\omega_k.$$
Since $G_0^*(\omega_i)=\omega_i$, we have $\Omega_i(0)=0$. We are going
to prove that $\Omega_i(t)\equiv 0$; this will imply that $G_t^*(\omega_i)$
is a linear combination of $\omega_1,\ldots,\omega_k$ for all $t$ and that
$X$ is indeed a symmetry of the distribution $P$. We have:
$${d\over dt}\Omega_i(t)=
    G_t^*(L_X\omega_i)\wedge\omega_1\wedge\ldots\wedge\omega_k
     = \sum_{j=1}^k G_t^*(\beta_{ij})\Omega_j(t),$$
which means that the vector consisting of $(k+1)$-forms
$\bigl(\Omega_1(t),\ldots,\Omega_k(t)\bigr)$,
is a solution to a linear homogeneous
system of ordinary differential equations with zero initial conditions.
Hence, it must vanish identically.

    This completes the proof of the theorem.
\medskip
    {\bf Example 5.} A simple computation using either of conditions 2
or 3 of the theorem, shows that vector field
$X=y\thinspace\der_y+\sin\varphi\thinspace\der_\varphi$ is a symmetry of the
oricycle distribution (see Example 2 above).

\beginsection 2. Symmetries and integration

The problem of {\sl integration} of a distribution consists in describing
its maximal integral manifolds. For a completely integrable distribution,
this is equivalent to finding a {\sl complete set of first integrals}.
A function $h\in C^\infty(M)$ is called a first integral of the
distribution $P$ if every integral manifold of $P$ lies entirely in
some level surface $\{x\in M\mid h(x)={\rm const}\}$ of this function, or,
equivalently, if $V(h)=0$ for any $V\in D(P)$.
 A complete set of first integrals
of the distribution $P$ is a set of functions whose mutual level surfaces
$$\{x \mid h_1(x)=c_1,\ldots,h_k(x)=c_k\}$$
represent the set of all maximal integral manifolds of $P$.

If $X$ is a symmetry with the flow $\{G_t\}$ and $Q$ a maximal integral
manifold of the distribution~$P$, then $G_t(Q)$  is also a maximal
integral manifold for any $t$. That is, the 1-parameter transformation group
generated by a symmetry preserves
the set of maximal integral manifolds (but possibly rearranges them in
some other order).
There is, however, a distinguished class of symmetries which
leave invariant every particular maximal integral manifold. These are
called {\sl characteristic symmetries}. By definition,
a symmetry of the distribution $P$ is said to be {\sl characteristic}
if it lies in $P$ (or, more exactly, belongs to~$D(P)$).
Transformations corresponding to a characteristic symmetry
move every maximal integral manifold along itself.
\medskip
{\bf Example.} The symmetry mentioned above in Example 5 is characteristic
while those given in Example 4 are not.
\medskip
The classical theorem of Cartan [St] implies that
the set of all characteristic symmetries
$$ \Char(P)=\Sym(P)\cap D(P)$$
is an ideal of the Lie algebra $\Sym(P)$. Elements of the quotient algebra
$$ \Shuf(P) = \Sym(P)/\Char(P)$$
will be referred to as {\sl shuffling symmetries} of the distribution $P$.
Flows corresponding to different representatives of a class
$$\bar X = X \bmod \Char(P)$$
rearrange ({\sl shuffle}) the set of maximal integral manifolds of $P$ in the
same way.

 Now suppose that $k=\codim P$ and $\G\subset \Shuf(P)$ is
a $k$-dimensional Lie subalgebra which is {\sl transversal} to the
distribution in the sense that the natural mapping
$\pi_x: \Shuf(P)\rightarrow T_x(M)/P_x$ is bijective at every point $x\in M$.
Let ${\bar X}_1,\ldots,{\bar X}_k$, where
  $${\bar X}_i=X_i \bmod \Char(P),\thinspace X_i \in \Sym(P),$$
 be a basis of $\G$ while
$\omega_1,\ldots,\omega_k$ be a basis of the $C^\infty (M)$-module
$\Lambda(P)$. The transversality condition for the algebra $\G$ is
equivalent to the requirement that the matrix
$$\Xi=\|\omega_i(X_j)\|\eqnocount$$
be nondegenerate at any point of the manifold. Hence one may choose another
basis $\omega_1',\ldots,\omega_k'$ of the module $\Lambda(P)$ in such a way
that the corresponding matrix $\|\omega_i'(X_j)\|$ would be the unit matrix.
This can be achieved by setting
$$\pmatrix{\omega_1'\cr \vdots\cr \omega_k'\cr}
                = \Xi^{-1}\pmatrix{\omega_1\cr\vdots\cr\omega_k\cr}$$
Note that the values $\omega_i(X_j)$ do not depend on the choice of
representatives $X_j \in {\bar X}_j \in \Shuf(P)$.

\medskip
{\bf Theorem 2.} {\sl Let $P$ be a completely integrable distribution
defined by the set of 1-forms $\omega_1,\ldots,\omega_k$. Let
 ${\bar X}_1,\ldots,{\bar X}_k$ be the basis of an algebra of
shuffling symmetries $\G \subset \Shuf(P)$. Suppose that
$\omega_i(X_j)=\delta_{ij}$ and
 $$[{\bar X}_i,{\bar X}_j]=\sum_{s=1}^k c_{ij}^s {\bar X}_s,$$
where $c_{ij}^s \in {\bf R}$. Then
 $$ d\omega_s=-{1\over 2}\sum_{i,j} c_{ij}^s \omega_i\wedge\omega_j. $$
}  % end of \sl instruction
\medskip
{\bf Proof.} By Frobenius' theorem,
$d\omega_s=\sum_j \gamma_{sj}\wedge\omega_j$, where $\gamma_{ij}$ are
appropriate 1-forms. Under the conditions specified, these 1-forms
actually belong to $\Lambda(P)$, i.e. vanish on the vectors from the
distribution. To see this, one should make use of the formula
$$L_{X_i}(\omega_s)=d(X_i\cchrg\omega_s)+X_i\cchrg d\omega_s.$$
According to Theorem 1, $L_{X_i}(\omega_s)\in\Lambda(P)$. Besides, by
the premises of the present theorem, $d(X_i\cchrg\omega_s)=0$ for all $i,s$.
Hence $X_i\cchrg d\omega_s=\sum_j \gamma_{sj}(X_i)\omega_j - \gamma_{ij}
\in \Lambda(P)$, which implies that $\gamma_{ij}\in\Lambda(P)$ and therefore
$$ d\omega_s= \sum_{i<j} \alpha_{ij}^s \omega_i\wedge\omega_j $$
for some appropriate functions $\alpha_{ij}\in C^\infty(M)$. It follows that
$$d\omega_s(X_i,X_j)=\alpha_{ij}^s,$$
for $i<j$. On the other hand,
$$d\omega_s(X_i,X_j) =X_i(\omega_s(X_j))-X_j(\omega_s(X_i))-\omega_s([X_i,X_j])
                     =-\omega_s(\sum_{r=1}^k c_{ij}^r X_r) = -c_{ij}^s.$$
Therefore $\alpha_{ij}^s=-c_{ij}^s$ and
$$ d\omega_s=-\sum_{i<j} c_{ij}^s \omega_i\wedge\omega_j
            =-{1\over 2}\sum_{i,j} c_{ij}^s \omega_i\wedge\omega_j.$$
\medskip
{\bf Corollary.} {\sl If the algebra $\G$ is commutative, then in the
conditions of the theorem a complete set
of first integrals
of the distribution can be found by quadratures.}
\medskip
Indeed, in this case all the forms $\omega_i$ are closed and thus locally
exact: $\omega_i=dh_i$ for some smooth functions $h_1,\ldots,h_k$. These
functions can be recovered by computing the integrals
$$ h_i(a)= \int_{a_0}^a \omega_i, $$
where $a_0$ is a fixed point of the manifold $M$.
\medskip
{\bf Remark.} As we mentioned in the introduction,
we use the words `integration by quadratures' in a
somewhat vague sense. When we say that something can be found
by quadratures, we mean that this something belongs to a suitable Liouville
type extension of the basic field and there is an algorithmic procedure to
recover it using known data. However, a notion of Liouville
type differential extensions which is wide enough to incorporate
finding implicit functions in the nonlinear case,
is still awaiting its precise definition.
\medskip
{\bf Example 6.} Let us find a first integral of the oricycle distribution
of Example 2 using its symmetry written down in Example 4 and in the
last end explain the origin of this title to an inquisitive reader.

We have $\omega=2\sin^2\negthinspace{\varphi\over 2}\thinspace dx
             +\sin\varphi\thinspace dy-y\thinspace d\varphi$ and
$X=\der_x$. The pairing $\omega(X)=2\sin^2{\varphi\over 2}$ is non-zero,
so we can take another basic 1-form
$$\omega'={1\over2\sin^2{\varphi\over 2}}\omega
=dx+\cot{\varphi\over2}dy-{1\over2\sin^2{\varphi\over2}}d\varphi,$$
which must be exact, according to the theorem. And it really is. In fact,
$$\omega'=d\bigl(x+y\cot{\varphi\over2}\bigr).$$

We see that the meaning of symmetries in this context is that they allow
one to find the integrating factor for a Pfaffian equation.

Now about the sense of the term `oricycle'. Consider the upper half-plane
$$H=\{x,y\mid y>0\}$$ as a model of the Lobachevsky geometry. The role of
straight
lines in this model is played by semi-circles perpendicular to the
$x$-axis (the `absolute'). Let~$M$ be the unitary tangent manifold of~$H$,
i.e. the set of all unit tangent vectors. As the third coordinate~$\varphi$
on the 3-dimensional manifold~$M$, we will
take the angle between the upward vertical direction and the given vector
(see Fig.~2). A simple calculation shows that a Lobachevsky line issuing
from the point $(x,y)$ at angle
$\varphi$, arrives at the point of the `absolute' with
coordinate $x+y\cot{\varphi\over2}$. This expression is the above found
first integral of the distribution under study. It remains to note that
the set of all vectors tending to the same point of the `absolute', is
called `oricycle' in hyperbolic geometry.
\medskip

Now suppose that we know a transversal algebra $\G$ of shuffling symmetries
of a distribution $P$, which is not commutative. We will show that if this
algebra is {\sl solvable} then it is possible to decompose the problem of
finding a complete set of first integrals for $P$ into a finite number of
steps, every one of which matches the assumptions of the Corollary above.

Denote the commutator subalgebra of $\G$ by $\G^{(1)}$ and suppose that
$\G^{(1)}\not=\G$. Then one can choose a basis
$\bar X_1,\ldots,\bar X_k$ of $\G$ in such a way that
$\bar X_1,\ldots,\bar X_r \not\in \G^{(1)}$ while
$\bar X_{r+1},\ldots,\bar X_k \in\G^{(1)}$. In this case
$c_{ij}^s=0$ for all $i,j$, if $s \leq r$. Choose the basis
$\omega_1,\ldots,\omega_k$ of $\Lambda(P)$ in such a way that
$\omega_i(X_j)=\delta_{ij}$. Theorem 2 implies that
1-forms $\omega_1,\ldots,\omega_r$ are closed and hence in some open
domain $\omega_1=dh_1$, \dots, $\omega_r=dh_r$. For an arbitrary choice
of constants $c=(c_1,\ldots,c_r)$ the level surface
        $$H_c=\{h_1=c_1,\ldots,h_r=c_r\}$$
is invariant under the commutator subalgebra $\G^{(1)}$, since
$X_j(h_i)=\omega_i(X_j)=0$ if $j\geq r+1$, $i \leq r$.

Let $P_c$ be the restriction of distribution $P$ to the surface $H_c$.
Distribution $P_c$ is completely integrable --- the foliation of $M$ whose
leaves are maximal integral manifolds of $P$ cuts a foliation
on the surface $H_c$. The dimension of $P_c$ is equal to the dimension
of $P$ while its codimension is $k-r = \dim\G^{(1)}$.

Observe that the restriction $\G^{(1)}|_{H_c}$ constitutes
a transversal algebra of shuffling symmetries of the distribution $P_c$.
Indeed, a shift along the trajectories of any field $X$ such that
$\bar X \in \G^{(1)}$ shuffles the leaves
of distribution~$P$ and preserves the manifold~$H_c$; hence it must
also shuffle the leaves of distribution~$P_c$. Transversality of
$\G^{(1)}$ follows from the fact that $\|\omega_i(X_j)\|$, $r<i,j\leq k$,
is the unit matrix, due to the above assumptions.

We can subject the pair $(P_c,\thinspace \G^{(1)}|_{H_c})$ to the same
procedure which was formerly applied to the pair $(P,\thinspace\G)$.
More precisely: let $\G^{(2)}=[\G^{(1)},\G^{(1)}]$ be the commutant of
$\G^{(1)}$. If $\G^{(2)} \not= \G^{(1)}$, then some of the 1-forms
$\omega_{r+1},\ldots,\omega_k$ are closed and give rise to local first
integrals of the distribution $P_c$. Distribution $P_c$ can be restricted
to the mutual level surface of these integrals, etc.

Now suppose that the algebra $\G$ is {\sl solvable}, i.e. the sequence
$\G \supset \G^{(1)} \supset \G^{(2)} \supset \ldots$, where
$\G^{(i+1)}=[\G^{(i)},\G^{(i)}]$, becomes zero
after a finite number of steps. Then the above described procedure
sooner or later will pose one into the conditions of the Corollary of
Theorem 2. Whence:
\medskip
{\bf Theorem 3.} {\sl Let $P$ be a distribution of codimension $k$.
Suppose that a solvable $k$-dimensional algebra of shuffling symmetries
of $P$, transversal to $P$, is known explicitely. Then $P$ is integrable by
quadratures, i.e. one can find a complete set of first integrals for $P$
by integrating closed 1-forms and solving functional equations.}
\medskip
    By solution of {\sl functional equations} (called so to distinguish
them from differential equations) we mean finding
functions defined by implicit formulas
$$F(z_1,z_2,\ldots,z_n)=0\Rightarrow z_1=f(z_2,\ldots,z_n).$$
    We will show how this theorem works on examples from the area of
differential equations. But first we will explain the connection
between equations and distributions.

\beginsection 3. Equations as distributions

In this section we will show how the above described algorithm
can be applied to ordinary differential equations, giving the
`second integration procedure' of S.~Lie (see [Sph] for a more
traditional exposition).

The study of an ordinary differential equation
$$F(x,y,y',\ldots,y^{(k)})=0\eqnocount$$
can be reduced to the study of a distribution in the following way.

Let $J^k\R$ be the manifold of $k$-jets of smooth functions on the line $\R$.
By definition, this is a space whose points correspond to all conceivable
sets of values taken by the independent variable $x$ (coordinate
in $\R$), dependent variable $y$ and the derivatives of the latter with
respect to the former up to order $k$. Hence, $J^k\R$ is a
$(k+2)$-dimensional space $\R^{k+2}$ whose coordinates can be designated by
$x$, $y$, $p_1$, $\ldots$, $p_k$.

Let $\C$ be the distribution in $J^k\R$ defined by
the set of 1-forms
$$\eqalign{ \omega_0     &= dy   - p_1dx,        \cr
            \omega_1     &= dp_1 - p_2dx,        \cr
                         &\ldots                 \cr
            \omega_{k-2} &= dp_{k-2}- p_{k-1}dx, \cr
            \omega_{k-1} &= dp_{k-1}- p_kdx.     \cr}\eqnocount$$
Following [KLV], we will call $\C$ the {\sl Cartan's distribution}.
A particular case of this object appeared earlier as Example 1.

The characteristic property of Cartan's distribution, which shows
its importance in the theory of differential equations, consists in the
following. A curve in $J^k\R$ that projects on the $x$-axis without
degeneration, is integral for $\C$ if and only if it has the form
$$y=y(x),\ p_1=y'(x),\ldots,\ p_k=y^{(k)}(x),\eqnocount$$
where $y(x)$ is a smooth function of $x$. Apart from these, there are
maximal integral manifolds of the second kind, namely straight lines parallel
to the $p_k$ axis. A general maximal integral manifold may include portions
of both types, fitted together to make a smooth curve (see [KLV] for the proof).

\ref=\formno \advance\ref by -2
Equation (\the\ref) corresponds to a hypersurface  $\E\subset J^k\R$,
defined by
$$F(x,y,p_1,\ldots,p_{k})=0,$$
The 2-dimensional Cartan's distribution $\C$, when restricted to $\E$,
produces a 1-dimensional distribution $\C_\E$, which will be referred
to as the {\sl Cartan's
distribution of the equation} $\E$. Maximal integral manifolds of this
distribution can contain segments parallel to the $p_k$ axis as well as
portions
of curves (\the\formno), where $y(x)$ must be a solution to equation
(\the\ref) in order that the curve should lie on $\E$. These pieces
are put together in the points where the projection to the $x$-axis
degenerates, and such points may constitute only a discrete set on the curve.
\medskip
{\bf Example 7.} Consider the equation
                    $$yy'+x=0.$$
We can take the variables $y$ and $p=y'$ for coordinates on the corresponding
surface $\E\subset J^1\R$. In these coordinate system
the distribution $\C_\E$ is given
by $(1+p^2)dy+yp\,dp=0$. Its integral curves $y=C(1+p^2)^{-1/2}$ are
depicted in Fig. 3a. Every pair of symmetrical curves produces one of the
circles in the $(x,y)$-plane which correspond to two-valued solutions
of the equation and are shown in Fig. 3b. Besides, there is a singular
integral curve $y=0$ on $\E$ which corresponds to the origin point
in the $(x,y)$-plane and gives rise to no solution.
\medskip
A vigilant reader might have noticed a flaw in our previous argument:
there can exist such points $x\in\E$ where
the plane of the Cartan's distribution is contained in $T_x\E$ and thus
$\dim \C_\E = 2$ at $x$. In this case $\C_\E$ is no distribution at all.
The study of such singular points requires special analysis. We will avoid it
by confining ourselves from now on to equations resolved with
respect to the highest derivative
$$y^{(k)}=f(x,y,y',\ldots,y^{(k-1)}).\eqnocount$$
It should be noted, however, that a major part of our considerations can be,
{\it mutatis mutandis}, transferred to the general case.

For an equation of type (\the\formno) the integral manifolds of the Cartan's
distribution can only consist of portions of type (4), so that
the problem of finding the (multivalued) solutions of such an equation
is equivalent to that of finding
integral manifolds of the distribution $\C_\E$. This distribution is
1-dimensional and hence completely integrable. Its codimension is equal to
the order $k$ of the equation under study. Theorem 3 applied in this
environment gives rise to the following
\medskip
{\bf Theorem 4.} {\sl If one knows explicitely a solvable $k$-dimensional
transversal
Lie algebra of symmetries of an ordinary differential equation of order $k$,
then one can find the general solution of this equation by quadratures.}
\medskip

By {\sl symmetries of the differential equation} we understand {\sl shuffling}
symmetries of its Cartan's distribution, i.e. elements of the quotient space
$$\Sym(\E)=\Shuf(\C_\E)=\Sym(\C_\E)/\Char(\C_\E).$$
Elements of this space have a convenient description in terms of the so called
{\sl generating functions\/} ([KLV],[Ib]). After introducing these, we will explain what does
the notion of transversality mean in terms of generating functions.

Let us view $x$, $y=p_0$, $p_1$, $\ldots$, $p_{k-1}$ as coordinates on the
hypersurface $\E$. In this coordinate system any vector field is written as
$$X=\alpha\dd{} x+\beta_0\dd{} {p_0}
                     +\ldots+\beta_{k-1}\dd{} {p_{k-1}}.$$
Note that the $C^\infty(M)$-module of characteristic symmetries of $\C_\E$
is generated by the field
$$D=\dd{} x+p_1\dd{} {p_0} +\ldots
    +p_{k-1}\dd{} {p_{k-2}}+f\dd{} {p_{k-1}},\eqnocount$$
the {\sl total derivative operator} with respect to $x$ on the equation $\E$.
Hence, in the quotient algebra $\Sym(\E)$ the following relation holds:
$$\dd{} x\equiv -p_1\dd{} {p_0} -\ldots
    -p_{k-1}\dd{} {p_{k-2}}-f\dd{} {p_{k-1}} \bmod \Char(\C_\E),$$
and it is sufficient to search symmetries only among vector
fields of the form
$$X=\beta_0\dd{} {p_0}
                     +\ldots+\beta_{k-1}\dd{}{p_{k-1}}.$$
Suppose that such a field $X$ is a symmetry of the Cartan's distribution.
In virtue of theorem 1, this means that the 1-forms $L_X(\omega_i)$,
$0\leq i<k$, belong to $\Lambda(\C_\E)$, i.e. they are linear combinations
 of 1-forms $\omega_0$, $\omega_1$, $\ldots$,
$\omega_{k-1}$ with functional coefficients.
Computing these Lie derivatives for $i<k-1$, we obtain
$$L_X(\omega_i)=(D(\beta_i)-\beta_{i+1})\thinspace dx
                               \thinspace\bmod \Lambda(\C_\E).$$
The latter 1-form is in  $\Lambda(\C_\E)$ if and only if
$\beta_{i+1} = D(\beta_i)$. Denoting $\beta_0$ by $\gf$, we arrive at
the following expression for $X$:
$$X=X_\gf=\sum_{i=0}^{k-1}D^i(\gf)\dd{} {p_i}.
                                     \eqnocount$$
In particular, we see that the field $X$ is entirely defined by one function
$\gf$, which is equal to the value of $X$ on $y$. This function $\gf$ will
be referred to as the {\sl generating function} of the vector field
$X_\gf$.

The space of Lie vector fields is thus isomorphic to the space
of smooth functions in $x,p_0,\dots,p_{k-1}$ and
we only have to transfer the commutator operation from the Lie algebra
of vector fields to the space of functions.
The resulting operation is given by the Poisson type formula
$$\{\gf,\gps\}=
\sum_{i=0}^{k-1}D^i(\gf)\dd{\gps} {p_i} -
\sum_{i=0}^{k-1}D^i(\gps)\dd{\gf} {p_i}$$
and will be exploited later in section 5.

The generating function of a symmetry of a given differential equation cannot
be arbitrary. The condition $L_X(\omega_{k-1})\in\Lambda(\C_\E)$ which can be
rewritten as
$$\bigl(D^k(\gf)-\sum_{i=0}^{k-1}D^i(\gf)
         \dd f {p_i}\bigr)dx \equiv 0 \bmod \Lambda(\C_\E)$$
implies that $\gf$ must satisfy differential equation
$\Delta(\gf)=0$, where the operator $\Delta$ ranging in the space
of smooth functions in $x$, $p_0$, \dots, $p_{k-1}$ is defined by
$$\Delta=D^k-\sum_{i=0}^{k-1}
           \dd f{p_i} D^i $$
(in the terminology of [KLV], operator $\Delta$ is the
{\sl universal linearization} of the function $F=p_k-f$ restricted to the
equation $\E$).

The discussion we have just finished can be summarized as follows.
\medskip
{\bf Theorem 5.}  $\Sym(\E)\cong\Ker\Delta$; {\sl this isomorphism is
effectuated by the correspondence $X_\gf\leftrightarrow\gf$, where
$X_\gf$ is defined by\/} (\prevref).
\medskip
Since evidently $\goo_i(X_\gf)=D^i(\gf)$ for the basic 1-forms (3) of
the Cartan's distribution, the pairing matrix (1) becomes
$\Xi=D^i(\gf_j)$, so that the algebra generated by $\gf_1,\ldots,\gf_k$
is transversal if and only if this matrix is non-degenerate.

Our approach to symmetries of differential equations is related to the
classical S.Lie's approach in the following way. Consider a vector field
$Y$ which is a symmetry
of the Cartan's distribution $\C$ in $J^k\R$. If $Y$ is tangent to the
equation $\E\in J^k\R$, then its restriction to $\E$ defines a symmetry of
the distribution $\C_\E$. In this case $Y$ is said to be an {\sl outer
symmetry} of the equation $\E$. Symmetries of the equation $\E$ in the sense
adopted above, i.e. symmetries of the distribution $\C_\E$, can be,
correspondingly, called {\sl inner symmetries}. Restrictions of outer
symmetries to $\E$ are distinguished from the set of all inner symmetries by
the property that their generating function depends only on $x$, $y$, $p_1$. This
follows from the theorem of Lie and B\"acklund: {\sl every symmetry
of the distribution $\C$ coincides with the natural lift to $J^k\R$
of some contact vector field in $J^1\R$}. The lifting procedure is described
in [IB, KLV, OL]; we shall not need its full description --- for our aims it
suffices to note that the contact field with generating function $\gf$ is
$$K_\gf=-\gf_p\dd{} x+(\gf-p\gf_p)\dd{} y + (\gf_x+p\gf_y)\dd{} p,$$
where $p=p_1$, and its lift to $J^k\R$ differs from the
right-hand part of this expression only in terms containing $\dh{}{p_2}$,
\dots, $\dh{}{p_k}$. Vector field $K_\gf$ coincides with the restriction of the
operator $X_\gf - \gf_p D$ to the space of functions in $x$, $y$, $p$.
Therefore, if $X_\gf$ is a symmetry of~$\E$, then both $X_\gf$ and the lift
of $K_\gf$ specify the same element of the quotient space $\Sym(\E)$.

    A particular class of contact fields is composed of lifts of the
so called infinitesimal {\sl point transformations}, i.e. infinitesimal
changes of independent and dependent variables or, in other words, vector
fields in the space $J^0\R$ with coordinates $x$, $y$.
Point transformations may be characterized by {\sl linear} dependence of their
generating functions on $p$. Indeed, according to (\the\formno), the
generating function of a symmetry $X$ is recovered by means of the formula
$\gf=\omega_0(X)$ and
$$\omega_0(\alpha\dd{} x + \beta\dd{} y +\ldots)=
                    \beta-\alpha p.$$
\medskip
{\bf Examples.} Here are point transformations which are most frequently
encountered in applications.

1. Translation in $x$ is the lift of the field $\dh{} x$ with generating function
$-p$.

2. Translation in $y$ is the lift of the field $\dh{} y$ with generating function
$1$.

3. Scale transformation is the lift of $ax\dh{} x+by\dh{}y,\ a,b\in\R$. Its
generating function is $by-apx$.
\medskip
{\bf Remark.} By commuting translations and scale transformations, one always
obtains translations, so that any Lie algebra consisting of vector fields of
these two kinds, is solvable.
\medskip

As the reader may see, the `inner' approach to symmetries is wider than the
`outer' approach which is confined only to contact transformations. From the
`outer' viewpoint, symmetries of the Cartan's distribution of an equation
$\E$ should be considered as `higher symmetries', since they are restrictions
to $\E$ of symmetries of the Cartan's distribution on the space of jets of
infinite order [KLV]. H.~Stephani [Sph] calls them `dynamic symmetries'.

However, all these `higher' symmetries of ordinary differential equations
can be reduced to mere point symmetries if one studies, instead of a single
equation, the corresponding first order system. We will explain the relevant
theory in the general context of finite type systems.
\beginsection 4. Finite type systems

Let us be given a system of differential equations
$$\eqalign {F_1(&x,u,\dd u x)=0,\cr
                &\dots          \cr
            F_r(&x,u,\dd u x)=0,\cr}\eqnocount$$
where $u$ and $x$ stand for the sets of dependent and independent variables
$u^1,\ldots,u^m$ and $x^1,\ldots,x^n$, respectively, while
$\dh u x$ designates the set of all derivatives of the former over
the latter up to a fixed order $k$.

The study of such system is geometrized like before, by considering
the submanifold~$\E$ in the jet space $J^k(n,m)$ equipped with the Cartan's
distribution. A natural system of coordinates in $J^k(n,m)$ is composed of all
$x^i$ for $i=1,\ldots,n$, all $u^j$ for $j=1,\ldots,m$ and variables
$u^j_\sigma$ where $1\leq j\leq m$ while $\sigma=i_1i_2\ldots i_n$ is a
multiindex with $|\sigma|=i_1+i_2+\cdots+i_n\leq k$. The Cartan's distribution
is defined by the set of 1-forms
$\omega^j_\sigma=du^j_\sigma-\sum_{i=1}^n u^j_{\sigma+(i)}dx^i$
for all $j=1,\ldots,m$ and $|\sigma|\leq k-1$, where $\gs+(i)$ is obtained
from $\gs$ by augmenting its $i$-th component by 1.
Solutions of equation\footnote*{\eightrm Here and below the word {\eightsl
equation} (singular) will be used as synonym to {\eightsl system of
equations}.}
(\the\formno) correspond to $n$-dimensional integral manifolds
of the restricted distribution $\C_\E$ whose projection to the
$(x^1,\ldots,x^n)$-plane is non-degenerate.

The above described general
notation for jet coordinates is rather cumbersome,
so in the situation when we have one unknown function
$u$ of two independent variables $x,y$, we will make use of Monge's notation
$$x,y,u,p,q,r,s,t$$
instead of $x^1,x^2,u^1,u^1_{10},u^1_{01},u^1_{20},u^1_{11},u^1_{02}$,
respectively.
\medskip
{\bf Example 8.} Consider the system
$$\eqalign{\dd u x &= {1-\cos u \over y},\cr
           \dd u y &= {\sin u \over y}   \cr}$$
in one unknown function $u$ of two independent variables $x$, $y$.
The relevant jet space $J^1(2,1)$ is 5-dimensional with coordinates
$x$, $y$, $u$, $p$, $q$. The equation under study specifies a
3-dimensional submanifold
$$\{p={1-\cos u\over y},\ q={\sin u\over y}\}$$
in this space, and the form of Cartan
                    $$du-pdx-qdy,$$
when restricted to this submanifold, becomes
         $$du-{1-\cos u \over y}dx - {\sin u \over y}dy,$$
which coincides, up to notations and a non-zero factor, with the 1-form
$\omega$ of Example 2, provided that $u$ is considered as a function
defined in the upper half of the $(x,y)$-plane and taking values modulo~$2\pi$.
It follows that our system has a 1-parametric family of solutions given by
maximal integral manifolds of the oricycle distribution.
\medskip
A system of differential equations is said to have {\sl finite type} if
its solution space is {\sl finite-dimensional}. Any system of ordinary
differential equations is obviously of finite type. In order that a system
of partial differential equations might have finite type, it normally needs
to be {\sl overdetermined}
in the sense that the number of equations in the system
should exceed the number of unknown functions.\footnote*{\eightrm This is
always true, e.g., when the system is {\eightsl involutive} (see [Ts], [Po1]).}

If the Cartan's distribution of an equation is $n$-dimensional and completely
integrable, as in Example 8, this equation is sure to have finite type.
However, this is not the only class of finite type systems.
\medskip
{\bf Example 9.} The system
           $$\eqalign {u_{xx}&=0\cr
                       u_{yy}&=0\cr}$$
is evidently of finite type, since its solutions are exausted by the
4-parametric family $u=axy+bx+cy+d$, $a,b,c,d\in\R$. However, the
corresponding geometric
image is a 6-dimensional plane $\{r=0,\ t=0\}$ in the 8-dimensional manifold
$J^2(2,1)$ whose Cartan distribution given by
$$\eqalign {  du-pdx-qdy&=0 \cr
              dp-sdy    &=0 \cr
              dq-sdx    &=0,\cr}$$
is 3-dimensional and not completely integrable.

If we consider the {\sl first prolongation} of this system
           $$\eqalign {u_{xxx}&=0\cr
                       u_{xxy}&=0\cr
                       u_{xyy}&=0\cr
                       u_{yyy}&=0,\cr}$$
obtained by taking total derivatives of all the equations of
the initial system, we will arrive at a 8-dimensional submanifold
(a plane, to be exact) in the jet space $J^3(2,1)\cong\R^{12}$
equipped with a 2-dimensional completely integrable distribution.
Thus, this example also can be studied in terms of completely integrable
distributions, but to effectuate this reduction, one first has to prolongate
the equation.
\medskip
The above definition of finite type systems is not constructive in the sense
that one should know the size of the space of solutions in order to decide
whether the system is of finite type. There is a class of systems
for which one can easily say that they are of finite type. We are speaking
 about systems
whose {\sl infinite prolongation} (see [KLV], [Ts]) is finite-dimensional.
This implies that after some number of prolongations the system can be resolved
with respect to all derivatives of a fixed highest order $k$. In this case all
variables $u^i_\gs$ with $|\gs|\geq k$ can be expressed through $x^j$, $u^s$ and
$u^s_\gt$ with $|\gt|<k$, so that the space $\A=C^\infty(J^{k-1})$ of smooth
functions on the manifold of $(k-1)$-jets serves as a closed universe for all
calculations related to the given system---much in the same way as the space
of functions in $x,y,y_1,\ldots,y_{k-1}$ did in the case of ordinary
differential equations.

In this space $\A$ the restrictions of total derivative operators
$$D_i=\dd{}{x_i} + \sum_{\gs,j} u^j_{\gs+(i)} \dd{}{u^j_\gs}$$
are defined. A generating function of a symmetry
is represented by a row
$(\gf^1,\ldots,\gf^m)\in\A^m$ , where $m$ is the number of dependent
variables $u^1,\ldots,u^m$.
The vector field in $J^{k-1}$ corresponding to this function is given by
the formula
$$X_{(\gf^1,\ldots,\gf^m)} = \sum_{\gs,j} D_\gs(\gf^j) \dd{}{u^j_\gs},$$
where $D_{i_1,\ldots,i_n}=D_1^{i_1}\circ\cdots\circ D_n^{i_n}$.
The generating function of a symmetry for
a system like (\the\formno) should satisfy $r$ equations
$$\sum_{\gs,j} \dd{F_i}{u^j_\gs} D_\gs(\gf^j) = 0, \ i=1,\ldots,r.$$
This constitutes a counterpart of Theorem 5 in the case of finite type systems.
All these formulas are valid for general systems of p.d.e.'s, but they have
to be considered in infinite-dimensional spaces
(see [KLV], [Ts] for proofs and details).
\medskip
{\bf Example.} The finite type system
$$\eqalign{u_x &= yv, \cr
            u_y &= xv, \cr
            v_x &= yu, \cr
            v_y &= xu \cr}$$
is described by a 4-dimensional surface with coordinates
$x=x^1$, $y=x^2$, $u=u^1$, $v=u^2$ in 8-dimensional jet space $J^1(2,2)$.
Below, we give the step-by-step records of the integration procedure
of Section 2 applied in this case (however, not explaining how the
symmetries $S_1$ and $S_2$ were found).


Basic forms of Cartan's distribution
$$\eqalign{\goo_1 &= du-yvdx-xvdy, \cr
            \goo_2 &= dv-yudx-xudy.  \cr}$$

Symmetries:
$$\eqalign{S_1 &= u\dd{}u+v\dd{}v, \cr
            S_2 &= e^{xy}(\dd{}u+\dd{}v). \cr}$$

Their generating functions:
$$\gf_1=\left(\matrix{u\cr v}\right),\
  \gf_2=\left(\matrix{e^{xy}\cr e^{xy}}\right).$$

Pairing matrix $S_i(\gf_j)$:
$$\Xi=\left(\matrix{u & e^{xy} \cr
                    v & e^{xy} \cr}\right).$$

New basic 1-forms:
$$\left(\matrix{\goo_1'\cr \goo_2'}\right)
 =\Xi^{-1} \left(\matrix{ du-yvdx-xvdy \cr dv-yudx-xudy}\right)
 =\left(\matrix{{d(u-v)\over u-v}+d(xy) \cr
       {-vdu+udv\over e^{xy}(u-v)} - e^{-xy}(u+v)d(xy)}\right)$$

First integrals:
$$\eqalign{&\goo'_1 = d[\log(u-v)+xy]\sim dh_1 , \ h_1=e^{xy}(u-v), \cr
&\goo'_2\big|_{h_1=\const} = d[e^{-xy}(u+v)] , \ h_2=e^{-xy}(u+v).\cr}$$

General solution:
$$\eqalign{u &= C_1e^{xy}+C_2e^{-xy}, \cr
            v &= C_1e^{xy}-C_2e^{-xy}.  }$$

\beginsection 5. More examples for o.d.e.'s

Here we give several examples of how the Theorem 4 can yield
explicit solutions for o.~d.~e.~'s.
Further examples can be found in an extensive literature that comprises,
e.g., the books [Ol], [Ovs], [Sph], [BK].
Note that the fundamental problem related to our method
is to {\sl find\/} enough symmetries of a given differential
equation in order that the integration procedure could work.
In Example 12, we show how one does this by equating coefficients
in the case of point symmetries.
This technique  is already quite traditional  and is
described here for the sake of the readers who are not familiar
with the literature.
One should note, however, that in recent years many computer
programs have appeared that enable one to search the symmetries
interactively. Among these, we would like to mention
the well-known REDUCE package by F.~Schwarz and the DELiA program
referred to below in the Acknowledgements.

\indent {\bf Example 10.} Let $\E$ be the linear equation
$$\alpha_k y^{(k)}+\cdots+\alpha_1 y'+\alpha_0 y=g,$$
where $\alpha_k,\ldots,\alpha_1,\alpha_0,g$ are given functions of $x$.
An arbitrary solution of the corresponding homogeneous equation
$$\alpha_k y^{(k)}+\cdots+\alpha_1 y'+\alpha_0 y=0$$
is a symmetry (generating function of a symmetry, to be precise) of the
initial equation. The space of all solutions makes a $k$-dimensional
commutative algebra, and if one knows its basis, i.e. a fundamental system
of solutions of the homogeneous equation, then the inhomogeneous equation
can be integrated by means of quadratures.

The computations needed to actually carry out the integration in this case,
are precisely the same which one comes accross when applying the usual trick
of `variation of constants'. In particular, the matrix $\Xi$ here consists
of functions of $x$ and hence is nothing but the usual Wronski matrix of the
fundamental system involved.
\medskip

{\bf Example 11.} In the well-known reference book on ordinary differential
equations [Ka] one can find the following equations:
$$\eqalign{4y^2y'''-18yy'y''+15y'^3 &=0, \cr
           9y^2y'''-45yy'y''+40y'^3 &=0, \cr}$$
which are treated as separate examples (No. 7.8 and 7.9). For each of them,
a separate %prescription
solution procedure is recommended. However, it is readily seen that both
equations as well as the arbitrary equation of the form
$$ay^2y'''+byy'y''+cy'^3 =0,\ a,b,c \in \R,\eqnocount$$
possess a 3-dimensional solvable Lie algebra of symmetries consisting of
the translation in $x$ and two independent scale transformations. It is
convenient to take the functions $\gf_1=y$, $\gf_2=p$, $\gf_3=y+xp$
for a basis of the symmetry algebra and follow
the above described scheme to accomplish the integration.

{\bf Remark.} This equation can be integrated in three steps by
succesively lowering its order, using first the translation invariance
and then invariance with respect to the two scale transformations. Thus, the
procedure of Lie and Cartan is a generalisation of standard methods of order
reduction. Note that the sequence in which the symmetries are used in this
procedure should conform to the Lie algebra structure of the given set of
symmetries. For example, if one tries to use scale invariance
of equation (\the\formno) first, then
the translation symmetry will be lost. In general, on the first step the
symmetries belonging to the last (smallest) commutator subalgebra should be
used.
\medskip
{\bf Example 12.} This one is the longest example in all the paper---
it even incorporates a theorem.
In compensation, an industrious reader will have an opportunity to
see in full detail how all the machinery works and what kind of results one
is supposed to obtain.
We will find all equations of the form
$$y''=y'+f(y), \eqnocount$$
which possess a two-dimensional Lie algebra of point symmetries and then
find the explicit expression for solutions of these equations in
terms of quadratures.

The independent variable $x$ does not enter explicitely into the equation,
hence the $x$-translation is a symmetry. The problem is to determine when
this equation has a second symmetry with generating function of the form
$$\gf=\alpha p + \beta, \eqnocount$$
where $p=p_1$ and $\alpha$ and $\beta$ are functions of $x$ and $y$ such
that $\{p,\gf\}$ is a linear combination of $p$ and $\gf$. In what follows,
we exclude the trivial particular case when the function $f(y)$ is linear.

We will use variables $x$, $y$ and $p$ as a system of coordinates on the
surface $\E\in J^2(1,1)$ corresponding to the given equation. In these
coordinates the total derivative operator has the form
$$D=\dd{}x + p\dd{}y + (p+f)\dd{}p. \eqnocount$$
The generating function of a symmetry $\gf(x,y,p)$ has to satisfy the
equation
$$D^2(\gf)-D(\gf)-f'\gf=0.$$
Taking into account relations (\prevref) and (\the\formno),
we can rewrite the last one as
$$\ga_{yy} p^2 +(2\ga_{xy}+2\ga_y+\gb_{yy})p^2+
   (3\ga_yf +\ga_x+\ga_{xx}+2\gb_{xy})p+
   (2\ga_x f +\gb_yf+\gb_{xx} -\gb_x -f'\gb)=0,$$
which is equivalent to the system of equations
$$\eqalign{&\ga_{yy} =0,                          \cr
           2&\ga_{xy}+2\ga_y+\gb_{yy} =0,         \cr
           3&\ga_yf +\ga_x+\ga_{xx}+2\gb_{xy} =0, \cr
           2&\ga_x f +\gb_yf+\gb_{xx} -\gb_x -f'\gb =0. \cr}\eqnocount$$

The first equation yields $\ga=\gg y+\gd$, where $\gg$ and $\gd$ are
some functions of $x$. Substituting this into the second equation, we obtain
$\gb=-(\gg+\gg')y^2+\ge y+\gz$, where $\ge$ and $\gz$ are again functions of
$x$. Then the third equation is reduced to
        $$3\gg f = 3(\gg+\gg')y-\gd'-\gd''-2\ge'.$$
Since the function $f(y)$ was supposed to be non-linear, it follows that
$\gg=0$ and $\ge=(d-\gd-\gd')/2$, where $d=\const$.

Now the last equation of the system (\the\formno) takes the form
$$(\ge y + \gz)f' - \gee f = \gth y + \gl \eqnocount$$
with $\gee=2\gd'+\ge$, $\gth = \ge''-\ge'$, which is an ordinary differential
equation with respect to the function $f(y)$, where the variable $x$ enters
as parameter. Its general solution, under the assumptions
$\ge\ne 0$, $\gee\ne 0$, $\ge\ne\gee$, is given by the formula
$$f=\gm\bigl(y+{\gz\over\ge}\bigr)^{\gee/\ge}+
{\gth\over\ge-\gee}\bigl(y+{\gz\over\ge}\bigr) +{\gth\gz-\ge\gl\over\ge\gee},
\eqnocount$$
where $\gm$ is an arbitrary function of $x$.

Among all functions (\the\formno) we have to choose those which depend only on $y$
and are nonlinear in $y$. The former requirement holds if and only if
all of the functions $\gm$, $\gz/\ge$, $\gee/\ge$, $\gth/(\ge-\gee)$,
$(\gth\gz-\ge\gl)/(\ge\gee)$ are constants. Denoting the constants
$\gm$, $\gz/\ge$, $\gee/\ge$ by $a$, $b$ and $c$ respectively and taking
into account all the relations among functions under consideration,
we arrive at the following expressions:
$$\eqalign{\ge  &= -{k+1\over2}ae^{kx}, \cr
           \gz  &= b\ge,                \cr
           \gee &= c\ge,                \cr
           \gth &= (k^2-k)\ge,          \cr
           \gl &= (k^2-k)b\ge,         \cr}$$
where $k={1-c\over c+3}$ (note that $c\ne-3$ if $\ge\ne0$). Hence
$$f(y)= a(y+b)^c - {2c+2\over(c+3)^2}y.$$
Now consider the possibilities previuosly excluded. Either of assumptions
$\gee=0$ and $\gee=\ge$ results in linearity of the function $f(y)$. In the
case $\ge=0$ we obtain a new series of solutions to equation (\prevref),
$$f(y)=ae^{by}-{2\over b},\ a,b\in\R.$$

The computations accomplished can be summarized as follows.
\medskip
{\bf Theorem 6.} {\sl Among all nonlinear equations of the form
$y''=y'+f(y)$ only equations of the two following series:
$$\leqalignno
{y'' &= y'+ a(y+b)^c- {2c+2\over(c+3)^2}y,\
                    a,b,c\in\R,\ c\ne-3, &({\rm A}) \cr
 y'' &= y'+ ae^{by}-{2\over b},\ a,b\in\R,\ b\ne 0 &({\rm B}) \cr}$$
possess a two-dimensional algebra of point symmetries. In case {\rm(A)}
a basis of this algebra may be chosen to consist of functions
$$\gf_1=p,\ \gf_2=e^{kx}(p-{k+1\over2}y)$$ with $k={1-c\over c+3}$,
in case {\rm(B)} the two functions $$\gf_1=p,\
\gf_2=e^{-x}(p-{2\over b})$$
can be taken as a basis.       }%end of \sl directive
\medskip
To complete the example, we will carry out the integration procedure
for the equation of series (A) with $a=1$, $b=0$, $c=-2$,
$$y''=y'+y^{-2}+2y.\eqnocount$$
The corresponding algebra of point symmetries is generated by
$\gf_1=p$ and $\gf_2=e^{3x}(p-2y)$. We have $\{\gf_1,\gf_2\}=-3\gf_2$.
The commutator subalgebra $\G^{(1)}$ is generated by $\gf_2$, hence the
basis $\gf_1$, $\gf_2$ is written exactly in the order prescribed by the
algorithm. The pairing matrix of symmetries and basic Cartan's 1-forms is
$$\Xi=\left(\matrix{\gf_1 &\gf_2 \cr
                    D\gf_1 &D\gf_2 \cr}\right)=
      \left(\matrix{p &e^{3x}(p-2y) \cr
                    q &e^{3x}(q+p-6y) \cr}\right),$$
where $q=p+y^{-2}+2y$. Its determinant is equal to $e^{3x}T$, where
$T=p^2-6yp+2yq = p^2-4yp+2y^{-1}+4y^2$. The new basic 1-forms are computed
as follows:
$$\left(\matrix{\goo'_0 \cr \goo'_1}\right)
  =\Xi^{-1}\left(\matrix{\goo_0 \cr \goo_1}\right)
  ={e^{-3x}\over T}\left(\matrix{e^{3x}(q+p-6y) &e^{3x}(2y-p) \cr
                                 -q &p}\right)
  \left(\matrix{dy-pdx \cr dp-qdx}\right),$$
whence
$$\eqalign{\goo'_0 &= {1\over T}[(2p+y^{-2}-4y)dy+(2y-p)dp]-dx,\cr
           \goo'_1 &= {1\over T}[-e^{-3x}(p+y^{-2}+2y)dy+e^{-3x}pdp].\cr}$$

The form $\goo'_0$ is closed; its integral is $-{1\over2}\log|T|-x$. Instead
of this function it is more convenient to take
$$h_1= \exp(-2\int\goo'_0)=e^{2x}(p^2-4yp+2y^{-1}+4y^2)$$
which is also a first integral of the equation.

Now we are to integrate the 1-form $\goo'_1$ on the surface $H_{c_1}$,
defined by $h_1=c_1=\const$.  On this surface
\def\sqrtone{\sqrt{c_1e^{-2x}-2y^{-1}}}
$$p=2y\pm\sqrtone$$
and hence
$$\goo'_1=\bigl(-e^{-3x}\mp{2ye^{-3x}\over\sqrtone}\bigr)dx
                 \pm {e^{-3x}\over\sqrtone}dy.$$
Computation of the integral of this 1-form depends on the sign of the
constant $c_1$. For $c_1>0$ we have
\def\sqrttwo{\sqrt{c_1-2y^{-1}e^{2x}}}
$$h_2={1\over3}e^{-3x}\pm{ye^{-2x}\over c_1}\sqrttwo
  \pm c_1^{-3/2}\log \Big|{\sqrttwo+\sqrt{c_1}\over\sqrttwo-\sqrt{c_1}}\Big|$$
and for $c_1<0$
$$h_2={1\over3}e^{-3x}\pm{ye^{-2x}\over c_1}\sqrttwo
     \mp (-c_1)^{-3/2}\arctan\sqrt{{2\over c_1}y^{-1}e^{2x}-1}.$$

The general solution to equation (\the\formno) is thus given by implicit formula
$h_2=c_2=\const$. Graphs of these solutions for all values of $c_1$ and
$c_2$ fill the region on the surface $\E$ where the algebra $\G$ is
transversal to the Cartan's distribution. The complement of this region
is defined by $\det\Xi=0$, i.e. $c_1=0$ or, explicitely,
$p=2y\pm\sqrt{-2y^{-1}}$. The latter relation can be interpreted as an ordinary
differential equation of first order whose solutions
$$y=-{1\over2}(Ce^{3x}\mp 1)^{2/3},\ C=\const,$$
are singular solutions of equation (\the\formno).
\medskip
{\bf Remark 1.} The famous MACSYMA system (Symbolics MacIvory, 1989 edition)
cannot solve equation (16).
\medskip
{\bf Remark 2.} One could from the very beginning reduce equation (\the\formno)
to a first order equation via the standard substitution $y'=z(y)$
(using translation invariance). However, this would
lead to the Abel's equation
$$z{dz\over dy}=z+y^{-2}+2y,\eqnocount$$
which is rather difficult to integrate. The reason is that (\the\formno) does not
inherit the other symmetry of (\prevref),
since the translation does not belong to the commutant subalgebra.
It seems that the simplest way to solve equation (\the\formno) consists in
passing over to second order equation (\prevref) (which {\it ipso facto} has
the translational symmetry), finding another invisible symmetry of the latter
and then integrating it with the help of the two symmetries---taken in
inverse order,
according to the structure of the Lie algebra. After doing this, one has to
filter the obtained two-parametric family of solutions of (\prevref)
thus chosing solutions of (\the\formno).

Of course, similar computations can be carried out for arbitrary values
of parameters $a$, $b$, and $c$ (see Theorem 6). Integration of equations
of type (A) reduces finally to integration of a differential binomial, and
Chebyshev's theorem implies that solutions are elementary functions whenever
$c=-{2\over n}-1$, where $n$ is integer---otherwise they are expressed through
elliptic integrals. For two specific values of parameter $c$, $c=2$ and $c=3$,
this
equation was solved by Painlev\'e [Pa], who, however, gave no hint as
to how he did that.
Equations of type (B) have elementary solutions for all $a$ and $b$.


\beginsection 6. Integration via overdetermination

It is well known that (systems of) partial differential equations (p.d.e.)
are very
reluctant towards explicit integration. Even finding one separate analytic
solution often should be considered as a big success. Following the line of our
previous argument, here we propose a method of searching for particular
solutions which is in a sense `perpendicular' to the standard procedure of
computing invariant, or automodel, solutions. However, the method is also
based on symmetries, and the equation under study should possess enough
symmetries in order that the method be applicable.

The idea is the following. Given a p.d.e. or a system which is not of finite
type but has an ample symmetry algebra, one should find complementary
equations in such a way that the resulting system would
have finite type and inherit as many
symmetries of the initial system as are necessary for the integration
procedure described in Section 2, i.e. in quantity equal to the
codimension of Cartan's distribution of the overdetermined system.

Since our aim is only to sketch a certain method,
in the subsequent considerations we will always mean generic situations,
assertions which hold on some open sets, or for almost all values of
parameters, etc., without explicitely mentioning that.
 We hope that the reader will excuse this liberty.

Let us discuss in more detail a possible way of hunting for
 particular solutions of one second order
equation imposed on the function $u(x,y)$.
In Monge's notation such an equation is written as
$$f(x,y,u,p,q,r,s,t)=0.\eqnocount$$
Suppose that we know a 4-dimensional solvable Lie algebra $\G$
of symmetries of this equation. We may try to find another equation
of the same kind
$$g(x,y,u,p,q,r,s,t)=0 \eqnocount $$
which is compatible with the first one and admits the same algebra $\G$.
Geometrically, the system (\prevref)-(\the\formno) represents a
6-dimensional submanifold
of the Monge's jet space $J^2(2,1)$. The relevant Cartan's distribution is
given by the set of 1-forms
$$\eqalign{&du-pdx-qdy\cr
           &dp-rdx-sdy\cr
           &dq-sdx-tdy\cr}\eqnocount$$
so that its restriction to the system is 3-dimensional, which does not meet
our needs, because the solutions we are looking for
should be represented by 2-dimensional surfaces.

However, after one prolongation the system will consist of 6 equations
so that the dimension of the corresponding submanifold of $J^3(2,1)$
will be 6.
If we choose the system of coordinates on this submanifold to be
$x,y,u,p,q,s$, then to obtain a basis of the Cartan's distribution, we
can add the 1-form $ds-u_{xxy}dx-u_{xyy}dy$ to the previously written
set, which means that the Cartan's distribution of the prolongated equation is
exactly 2-dimensional.

Let us try to run the entire procedure, finding explicitely a
4-parametric family
of solutions for the Monge-Amp\`ere equation
$$u_{xx}u_{yy}-u_{xy}^2=u^2,$$
or in Monge's notation,
                    $$rt-s^2=u^2.\eqnocount$$
This equation possesses a huge amount of symmetries. Already its algebra of
point symmetries has dimension 6. Let us take four point symmetries which
are most `visible', two translations with generating functions $\gf_1=p$ and
$\gf_2=q$ and two scale transformations with generating functions
$\gf_3=xp-yq$ and $\gf_4=u$. The lifts of the corresponding contact
vector fields to the space of 2-jets are
$$\eqalign{X_1 &=-\dd{}x, \cr
           X_2 &=-\dd{}y, \cr
           X_3 &=-x\dd{}x +y\dd{}y +p \dd{}p-q\dd{}q+2r\dd{}r-2t\dd{}t , \cr
           X_4 &= u\dd{}u+p\dd{}p+q\dd{}q+r\dd{}r+s\dd{}s+t\dd{}t , \cr}$$
and they form a solvable Lie algebra $\G$.

 The general second order equation invariant under this algebra is
$$F(u^{-2}pq,up^{-2}r,u^{-1}s,uq^{-2}t)=0,\eqnocount$$
$F$ being an arbitrary function of specified arguments.
One may try to determine all equations (\the\formno) compatible with equation
(\prevref) in the sense that they have a 4-parametric family
of common solutions.
This notion of {\sl complete compatibility\/} is equivalent to complete
integrability of the Cartan's distribution after the first prolongation
and can be handled in the following way.

In the generic situation the system (21)---(22) can be rewritten as
$$\eqalign{r &= {u(u^2+s^2)\over
                q^2\,f(u^{-2}pq,u^{-1}s,u^2p^{-2}q^{-2}(u^2+s^2))}, \cr
           t &= u^{-1}q^2\,f(u^{-2}pq,u^{-1}s,u^2p^{-2}q^{-2}(u^2+s^2)),\cr}$$
where $f$ is an arbitrary function of its arguments.
These equations should be once differentiated giving some expressions for
the third derivatives $K=u_{xxx}$, $L=u_{xxy}$, $M=u_{xyy}$, $N=u_{yyy}$
in terms of $x,y,u,p,q,s$.
All geometrical objects here and below are assumed to be defined on the
resulting 6-dimensional submanifold in $J^3(2,1)$.
We will use variables $x$, $y$, $u$, $p$, $q$ and $s$ as coordinates on this
manifold, understanding all the other involved letters as their functions.

The basic vector fields that span the
Cartan's distribution of the prolongated system are
$$\eqalign{X &= \dd{}x +p\dd{}u +r\dd{}p +s\dd{}q +L\dd{}s, \cr
           Y &= \dd{}y +q\dd{}u +s\dd{}p +t\dd{}q +M\dd{}s, \cr}$$
and the integrability condition $[X,Y]=\gl X+\gm Y$ is equivalent to
     $$X(M)=Y(L).\eqnocount$$
This is a first order nonlinear partial differential
equation imposed on the function $f$. Of course, it is rather
complicated (we do not even write it explicitly in terms of $f$!)
and the problem of finding its general solution is difficult.
But every particular solution of this equation gives rise to
a specific auxiliary equation (22) and thus to a 4-parametric
family of solutions to the initial equation (21).
One may try various ansatzes for the function $f$ in order to
find a solution of (24). We have tried $f=1$ and found out
that it just complies with the integrability condition (24).

Let us execute the integration procedure for the system
$$\eqalign{r &=us^2q^{-2} + u^3q^{-2}, \cr
           t &=u^{-1}q^2, \cr}\eqnocount$$
corresponding to $f=1$.
We will use the algebra $X_1,\ldots,X_4$ and the following set of
basic 1-forms for the Cartan's distribution:
$$\eqalign{\goo_1 &= du-pdx-qdy\cr
           \goo_2 &= dp-rdx-sdy\cr
           \goo_3 &= dq-sdx-tdy\cr
           \goo_4 &= ds-Ldx-Mdy,\cr}$$
where
$$\eqalign{L &= 3q^{-1}s^2 -2u^{-1}ps +q^{-1}u^2, \cr
           M &= 2u^{-1}qs - u^{-2}pq^2. \cr}$$

Since the commutator subalgebra of $\G$ is spanned by $X_1$ and $X_2$,
by theorem 2 we know that in the new basis of 1-forms
$\goo'_1,\ldots,\goo'_4$, such that $\goo'_i(X_j)=\gd_{ij}$, two
forms, $\goo'_3$ and $\goo'_4$, will be closed. A simple calculation yields
$$\goo'_3={1\over2uqs-2pq^2}[qsdu -q^2dp +(pq-2us)dp +uqds]$$
and
$$\int{\goo'_3}=-{1\over2}\log(us-pq)+\log q.$$

The function $h=(us-pq)/q^2$ is thus a first integral of the system (23).
According to the general scheme, we now have to integrate $\goo'_4$ and
then, restricting everything to the mutual level surfaces of the two first
integrals, compute the integrals of the two remaining basic 1-forms.
But profiting by the apparent simplicity of function $h$, we might as well
interrupt the smooth flow of the algorithm here and start it anew in a simpler
environment. Here is what we mean by that.

Existence of the first integral $h$ implies that
the system (23) is equivalent to a 1-parametric family of equations
$$\eqalign{r &=us^2q^{-2} + u^3q^{-2}, \cr
           s &=Cu^{-1}q^2 + u^{-1}pq, \cr
           t &=u^{-1}q^2 \cr}\eqnocount$$
for arbitrary $C\in\R$. For any specific value of $C$ equation (25)
represents
a 3-dimensional submanifold in $J^2(2,1)$ and the corresponding Cartan's
distribution is 2-dimensional, so that there is no need for prolongation and
everything can be done within a 5-dimensional manifold with coordinates
$x$, $y$, $u$, $p$, $q$, using the set of 1-forms (20) and
only three symmetries, say $X_1$, $X_2$ and $X_4$.
This algebra is commutative, so that all the three 1-forms of the new
recalculated
basic set are closed. Their integration gives the following first integrals:
$$\eqalign{& {au\over q}+x =\const, \cr
           & {a^2u^5\over5q^5} + {aup\over q^2} +{u\over q} - y = \const, \cr
           & {a^2u^4\over4q^4} +{ap\over q} - \log q = \const,\cr} $$
where $a=C^{-1}$ (which almost always makes sense!),
and after eliminating $p$ and $q$ we arrive at the explicit formula for
a 4-parametric family of solutions to the initial equation (21):
$$u={x+C_1\over a}\exp\bigl[{(x+C_1)^4\over20a^2}
          +{a(y+C_2)\over x+C_1} + C_3 \bigr].$$

\medskip
{\bf Remark.} One could try to handle the system (23) by first solving its
second equation. This way leads, however, to a rather intricate nonlinear
second order o.d.e. which the authors were not smart enough to cope with.
This is yet another confirmation of S.Lie's motto
$${\sl Use\ symmetries\ to\ solve\ differential\ equations!}$$

\beginsection Acknowledgements

One of the authors (S.D.) would like to acknowledge financial support of
the World Laboratory, which helped him to accomplish this work during his stay
at Ettore Majorana Centre, Sicily. A part of computations was carried out
using software systems DELiA of A. Bocharov and M. Bronstein
(Program Systems Institute) and CoCoA of A. Giovini and G. Niesi
(University of Genoa, {\tt cocoa@igecuniv.bitnet}).
Finally, we thank our referees for numerous beneficial remarks.

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