\setcounter{section}{5}%so that this will be section 6
\section{Concluding Remarks}\label{wconclusion}
In this section, we will indicate some interesting directions in which the results in
this paper might be extended, and some important questions about \BT/s to which the
techniques in this paper may be relevant.

\begin{enumerate}
\item  The set of equivalence classes (under contact transformations) of second-order \MA/ equations
to which the results of \S3 in this paper apply is relatively small, confined to the equations on
the Goursat-Vessiot list.  It would be interesting to see if the arguments in that section
could be applied to hyperbolic systems of class $k>1$.  In other words, given a hyperbolic system $\I$
of class $k$, linked to the standard wave equation by a \BT/, can one prove that the prolongation of $\I$ is
Darboux-integrable?
Likewise, given a hyperbolic EDS $\I$ of
class $k$, such that its prolongation is Darboux-integrable, does there exist a \BT/  between
$\I$ and the \MA/ system for the standard wave equation?  (The argument given at the end of
Chapter 7 in \cite{CFB} shows that there is a \BT/ between the wave equation and the {\em prolongation}
of $\I$; however, for practical purposes it is desirable to have
a \BT/ between systems of as low an order as possible, so that one has a smaller
system of ODE to solve in order to construct solutions.)  These hyperbolic systems would include, for example, the \MA/ equations which
are Darboux-integrable at third order, which have not been classified and are thought
to comprise a much larger set.

\item
It is a theorem of Sophus Lie that no \MA/ equation of the form $u_{xy}=f(u)$ is Darboux-integrable
(after arbitrary many prolongations) except when $f(u) = \exp(a u +b)$ for constants $a$ and $b$ (see \cite{GLecons}, Chapter IX).
Consequently, important equations like sine-Gordon cannot have a \BT/ to the wave equation.
Instead, the \BT/ \eqref{SGsingleBT} for sine-Gordon produces solutions to the {\em same} PDE
as we started with.  This is known as an {\em auto-\BT/};\footnote{This terminology is not universally accepted; Hongyou Wu \cite{Wu} has proposed
that transformations between different PDEs be known as {\em Miura transformations},
and the term \BT/ be reserved for what we are calling auto-\BT/s.}
 such transformations play an important
role in the theory of completely integrable PDE \cite{Fordy}.

It is therefore of interest to
try to identify those \MA/ equations which have non-trivial auto-\BT/s.  We remark that
for such transformations, the \MA/ systems $\I$ on $\scM$ and $\Ibar$ on $\Mbar$
must be contact-equivalent, i.e.,. there must be a diffeomorphism $\Phi: \scM \to \Mbar$ which
pulls back $\Ibar$ to $\I$.

\setlength{\unitlength}{2pt}
\begin{center}
\begin{picture}(40,30)(0,0)
\put(5,5){\makebox(0,0){$\scM$}}
\put(35,5){\makebox(0,0){$\Mbar$}}
\put(20,25){\makebox(0,0){$\B$}}
\put(16,21){\vector(-3,-4){9}}
\put(24,21){\vector(3,-4){9}}
\put(10,5){\vector(1,0){21}}
\put(18,7){$\Phi$}
\put(7,16){$\pi$}
\put(31,16){$\pibar$}
\end{picture}
\end{center}
Necessary conditions for the existence of such a diffeomorphism may be derived
from the fact that it is required to preserve the differential invariants of the \MA/ systems.
(See \cite{BGG}, \S2.1, for a derivation of these invariants using the method of equivalence.)

\item Our previous paper \cite{CI} began the exploration of {\em parametric} \BT/s using
the method of equivalence.   Such transformations contain an arbitrary parameter in the
\BB/ system; for example, an arbitrary nonzero parameter $\lambda$ may be interpolated
in the sine-Gordon auto-\BT/  \eqref{SGsingleBT} to give
$$
\begin{aligned}
v_x - u_x&=  \dfrac{\lambda}{2}\sin((u+v)/2), \\
v_y + u_y&=  -\dfrac{1}{2\lambda}\sin((u-v)/2).
\end{aligned}
$$
One observes that this system differs from \eqref{SGsingleBT} merely by scaling
$x$ by $\lambda$ and $y$ by $\lambda^{-1}$---a change of variables which is a
symmetry of the sine-Gordon equation but not of the system \eqref{SGsingleBT}.  This scaling symmetry
can also be applied to the \BT/ \eqref{Liouvillewave}, to produce a parametric transformation
$$
z_x = u_x - 2\lambda\exp((u+z)/2), \qquad
z_y = -u_y +\dfrac{1}{\lambda}\exp((u-z)/2),
$$
where $u(x,y)$ satisfies Liouville's equation and $z(x,y)$ solves the wave equation.
(In fact, this transformation is derived from the most general form \eqref{Liouvillegeneral}
by setting $k=1$ and choosing $v(x)=2\ln \lambda$ and $w(y)=0$.)  In \cite{CI} it is shown that
these transformations can be generated from a non-parametric \BT/  by starting with a symmetry vector field
on $\scM$, choosing a lift into $\B^6$ which is {\em not} a symmetry of the Pfaffian system
$\J$, but such that pulling $\J$ back to $\B \times \R$ via the 1-parameter
family of diffeomorphisms generated by the lift gives a family of transformations.
The same approach can be taken with other transformations discussed in \S4.1; for example,
the transformation \eqref{G2bt} may be generalized to a parametric transformation
$$z z_x - \lambda u u_x = \sqrt{z^2 - \lambda u^2}\sqrt{\lambda+z_x^2}, \qquad
z z_y - \lambda u u_y = \sqrt{z^2 - \lambda u^2}\sqrt{\lambda+z_y^2}
$$
for $\lambda >0$, where $u$ satisfies (II) and $z$ solves the wave equation.
(This is obtained by starting with the symmetry of (II) that simultaneously
scales $u$, $x$ and $y$.)

With these examples in evidence, and given the importance of parametric \BT/s
in the study of `soliton' equations, it is desirable to characterize those transformations
that may be made to depend on an arbitrary parameter by lifting symmetry vector fields.




\end{enumerate}
