\section{$G$-structure for \BT/s}\label{wsetup}

The material in this section is taken from the first author's paper \cite{C1}; additional details may be found there.

Suppose that $(\scM, \I)$ and $(\Mbar, \Ibar)$ are hyperbolic
Monge-Amp\`ere systems, with
\[ \I = \{ \theta, \Omega_1, \Omega_2 \}, \qquad \Ibar = \{ \thetabar, \Omegabar_1, \Omegabar_2 \}.\]
As a special case of Definition \ref{general-Backlund-defn}, we define a B\"acklund transformation between $(\scM, \I)$ and $(\Mbar,
\Ibar)$ to be a 6-dimensional submanifold $\B \subset \scM \times \Mbar$
for which the pullbacks to $\B$ of the forms
$\Omega_1, \Omega_2, \Omegabar_1, \Omegabar_2$ have the property that
\begin{equation*}
 \Omega_i \equiv \Omegabar_i \mod{\{\theta, \thetabar\}}, \ \ i=1,2.
\end{equation*}
(The vector bundles $J,\Jbar \subset T^*\scB$ mentioned in Definition \ref{general-Backlund-defn}
are in this case spanned by the pullbacks of $\thetabar$ and $\theta$, respectively.)
A \BT/ is {\em normal} in the sense of Theorem \ref{ourtheorem} if
the pullbacks to $\B$ of the 2-forms $d\theta$, $d\thetabar$ are linearly independent modulo $\{\theta, \thetabar\}$.

Now let $\J$ be the ideal on $\B$ generated by the pullbacks of $\I$ and $\Ibar$; according to the conditions above, $\J$ is generated algebraically by the forms $\{\theta, \thetabar, d\theta, d\thetabar \}. $

Since $\I$ and $\Ibar$ are hyperbolic, locally there exist 1-forms
$\om^1, \om^2,
\om^3, \om^4$ on $\B$ such that $\{\theta,\, \thetabar,\, \om^1,\,
\om^2,\, \om^3,\, \om^4\}$ is a coframing of $\B$ (i.e., a set of
1-forms that restricts, at each point, to be a basis for the cotangent
space of $\B$)  and
\[ \J = \{\theta,\, \thetabar,\, \om^1 \& \om^2,\, \om^3 \& \om^4\}.
\]
(It is important to note that $\theta$ and $\thetabar$ are each
{\em separately} determined up to a scalar multiple, since each
determines the contact structure on a 5-manifold.)
Any such coframing has the property that
\begin{align*}
d\theta & \equiv A_1\, \om^1 \& \om^2 + A_2\,\om^3 \& \om^4
\mod{\{\theta,\, \thetabar\}},\\
d\thetabar & \equiv A_3\, \om^1 \& \om^2 + A_4\, \om^3 \& \om^4
\mod{\{\theta,\, \thetabar\}}
\end{align*}
for some nonvanishing functions $A_1, A_2, A_3, A_4$.  Since $d\theta,\,
d\thetabar$ are required to be linearly independent 2-forms at each
point of $\B$, we must have $A_1 A_4 - A_2 A_3 \neq 0$.

By rescaling the $\om^i$ and adding multiples of $\theta$ and
$\thetabar$ to the $\om^i$ if necessary, we can arrange that
\begin{align}
d\theta & \equiv A_1\, \om^1 \& \om^2 + \om^3 \& \om^4
\mod{\theta}, \label{dtheta}\\
d\thetabar & \equiv \om^1 \& \om^2 + A_2\, \om^3 \& \om^4
\mod{\thetabar} \notag
\end{align}
for some nonvanishing functions $A_1,\, A_2$ on $\B$ with $A_1
A_2 \neq 1$.
This coframing is not unique; any other such coframing
$\{\tilde{\theta},\, \tilde{\thetabar},\, \tilde{\om}^1,\,
\tilde{\om}^2,\,
\tilde{\om}^3,\, \tilde{\om}^4\}$ has the form
\begin{equation}
 \begin{bmatrix} \tilde{\theta}\\[0.1in] \tilde{\thetabar}\\[0.1in]
\tilde{\om}^1\\[0.1in] \tilde{\om}^2\\[0.1in] \tilde{\om}^3\\[0.1in]
\tilde{\om}^4
\end{bmatrix} =   \begin{bmatrix} b_{11}b_{22} - b_{12}b_{21} & 0 &
0 & 0 & 0 & 0 \\[0.1in] 0 &  a_{11}a_{22} - a_{12}a_{21} & 0 & 0 & 0 & 0
\\[0.1in] 0 & 0 & a_{11} & a_{12} & 0 & 0 \\[0.1in] 0 & 0 & a_{21} &
a_{22} & 0 & 0
\\[0.1in] 0 & 0 & 0 & 0 & b_{11}&  b_{12} \\[0.1in] 0 & 0 & 0 &
0 & b_{21} & b_{22}
\end{bmatrix}^{-1} \begin{bmatrix} \theta\\[0.1in] \thetabar\\[0.1in]
\om^1\\[0.1in]
\om^2\\[0.1in] \om^3\\[0.1in] \om^4 \end{bmatrix},  \label{G0freedom}
\end{equation}
where
$ b_{11}b_{22} - b_{12}b_{21} \neq 0, \
a_{11}a_{22} - a_{12}a_{21} \neq 0.$  (The inverse is included for
greater ease of computation in carrying out the method of equivalence.)  A coframing satisfying
\eqref{dtheta} is called {\em adapted}, and the group $G$ of matrices
of the
above form is called the {\em structure group} of the equivalence
problem.  (In fact, the most general choice of structure group would
include a discrete component interchanging the
distributions $\{\om^1, \om^2\}$ and $\{\om^3, \om^4\}$.  However,
this freedom does not contribute anything crucial to the structure
group, and it is easier to work with a connected group.)  The associated {\em $G$-structure} is the principal $G$-bundle $\calP \to \B$  whose local sections are precisely the adapted coframings over a neighborhood of $\B$.

In \cite{C1}, it is shown that $\calP$ has {\em structure equations}
\begin{multline}
\begin{bmatrix} d\theta\\[0.1in] d\thetabar\\[0.1in] d\om^1\\[0.1in]
d\om^2\\[0.1in] d\om^3\\[0.1in] d\om^4
\end{bmatrix} =
-\begin{bmatrix}\bet_1 + \bet_4 & 0 & 0 & 0 & 0 & 0 \\[0.1in] 0 & \alp_1 +
\alp_4 & 0 & 0 & 0 & 0 \\[0.1in] 0 & 0 & \alp_1 & \alp_2 & 0 & 0
\\[0.1in] 0 & 0 &
\alp_3 & \alp_4 & 0 & 0 \\[0.1in] 0 & 0 & 0 & 0 & \bet_1 & \bet_2
\\[0.1in] 0 & 0 & 0 &
0 & \bet_3 & \bet_4 \end{bmatrix} \&
\begin{bmatrix} \theta\\[0.1in] \thetabar\\[0.1in] \om^1\\[0.1in]
\om^2\\[0.1in] \om^3\\[0.1in] \om^4
\end{bmatrix}  \\[0.2in] \label{G0struct}
+ \begin{bmatrix} \theta \& (A_1 C_2\, \om^1 - A_1 C_1\, \om^2) + A_1\, \om^1 \& \om^2 + \om^3 \& \om^4
\\[0.1in]
\thetabar \& (A_2 C_4\, \om^3 - A_2 C_3\, \om^4) + \om^1 \& \om^2 + A_2\, \om^3 \&
\om^4 \\[0.1in]
B_1\, \theta \& \thetabar + C_1\, \om^3 \& \om^4  \\[0.1in]
B_2\, \theta \& \thetabar + C_2\, \om^3 \& \om^4  \\[0.1in]
B_3\, \theta \& \thetabar + C_3\, \om^1 \& \om^2  \\[0.1in]
B_4\, \theta \& \thetabar + C_4\, \om^1 \& \om^2 \end{bmatrix}
\end{multline}
for some functions $A_i, B_i, C_i$ and 1-forms  $\alp_i, \bet_i$ on $\calP$.
These equations are chosen so
that the matrix in \eqref{G0struct} takes values in the Lie algebra
$\mathfrak{g}$
of $G$; this is a standard step in the method of equivalence.  (See
\cite{G89} for details.)

The 1-forms $\alp_i, \bet_i$ are linearly independent from each other and from $\theta, \thetabar, \om^i$; they are called {\em pseudoconnection forms,} or more concisely (but imprecisely) {\em connection forms} on $\calP$.  They are well-defined only up to transformations of the form
\begin{alignat}{2}
\alp_1 & \mapsto \alp_1 + r_1\, \om^1 + r_2\, \om^2, \qquad \qquad & \bet_1 &
\mapsto \bet_1 + s_1\, \om^3 + s_2\, \om^4, \notag \\
\alp_2 & \mapsto \alp_2 + r_2\, \om^1 + r_3\, \om^2, \qquad \qquad & \bet_2 &
\mapsto \bet_2 + s_2\, \om^3 + s_3\, \om^4, \label{G0connfreedom} \\
\alp_3 & \mapsto \alp_3 + r_4\, \om^1 - r_1\, \om^2, \qquad \qquad & \bet_3 &
\mapsto \bet_3 + s_4\, \om^3 - s_1\, \om^4, \notag \\
\alp_4 & \mapsto \alp_4 - r_1\, \om^1 - r_2\, \om^2, \qquad \qquad & \bet_4 &
\mapsto \bet_4 - s_1\, \om^3 - s_2\, \om^4.\notag
\end{alignat}

\begin{rem}
The coefficients $A_i, B_i, C_i$ are called {\em torsion functions}.  They may be
interpreted as the components of well-defined tensors associated to the \BT/, as follows.

A hyperbolic \MA/ system naturally equips the underlying manifold
$\scM^5$ with a line bundle $L \subset T^*\scM$ and
two rank 3 characteristic bundles $K_1, K_2 \subset T^*\scM$ whose intersection is $L$.
(The generator 1-form $\theta$ is a section of $L$, and the factors of the
decomposable generator 2-forms $\Omega_1$ and $\Omega_2$ span a complement of $L$ within $K_1$ and $K_2$,
respectively.)  The $G$-structure for the \BT/ shows that $\scB^6$ is equipped with
a well-defined splitting of its cotangent bundle:
\def\Lbar{\overline{L}}
\begin{equation}\label{TBsplit}
T^*\scB = L \oplus \Lbar \oplus W_1 \oplus W_2,
\end{equation}
where $L$ and $\Lbar$ are the pullbacks of the \MA/ line bundles from $\scM$ and $\Mbar$
respectively, and $W_1, W_2$ are spanned by $\w^1, \w^2$ and $\w^3, \w^4$ respectively.
The normal \BB/ condition implies that
$$\pi^* K_i = L \oplus W_i, \qquad \pibar^* \overline{K}_i = \Lbar \oplus W_i, \qquad i=1,2.$$

The splitting \eqref{TBsplit} induces a corresponding splitting of $\Lambda^2 T^* \scB$,
whose summands include the line bundles $L \otimes \Lbar$, $\Lambda^2 W_1$ and $\Lambda^2 W_2$.  We may then define
a natural map from sections of $L$ to sections of $\Lambda^2 W_1$, given by
$$\delta_0: \theta \mapsto \text{projection of } d\theta \text{ into } \Lambda^2 W_1.$$
But this map is linear under multiplication by functions, and so gives a well-defined
map between the corresponding vector bundles.
The structure equations \eqref{G0struct}
show that $A_1 \w^1 \& \w^2$ is the value of $\delta_0$ applied to the first
member $\theta$ of the coframe.  Hence, $A_1$ is the component, with respect to
the give coframe, of a well-defined tensor in $L^* \otimes \Lambda^2 W_1$.
Similarly, $A_2$ is the component of a well-defined tensor
in $\Lbar^* \otimes \Lambda^2 W_2$

We may similarly define a map on sections of $W_1$ by
$$\delta_1: \w \mapsto \text{projection of } d\w \text{ into } L_1 \otimes L_2,$$
which again is linear under multiplication by functions.  The structure equations
show that $B_1 \theta \wedge \thetabar$ and $B_2 \theta \wedge \thetabar$ give
the value of $\delta_1$ on the basis sections $\w^1, \w^2$ of $W_1$ respectively.
Thus, the vector $[B_1\ \ B_2]$ gives the components, with respect to the coframe, of a well-defined
tensor in $W_1^* \otimes L_1 \otimes L_2$.  In a similar way, we see that
$[B_3\ \ B_4]$ are the components of a tensor in $W_2^* \otimes L_1 \otimes L_2$,
$[C_1\ \ C_2]$ are the components of a tensor in $W_1^* \otimes \Lambda^2 W_2$,
and $[C_3\ \ C_4]$ are the components of a tensor in $W_2^* \otimes \Lambda^2 W_1$,
all defined by taking the exterior derivative of a section and projecting
into the appropriate summand of $\Lambda^2 T^* \scB$.
\end{rem}
\bigskip

The following results are proved in \cite{C1}:

\begin{prop}
The vectors $[B_1 \ \ B_2], \ [B_3 \ \ B_4], \ [C_1 \ \ C_2], \ [C_3 \ \ C_4]$ are {\em relative invariants:} given any point $m \in \B$, they are each either zero for every adapted coframing at $m$, or nonzero for every adapted coframing at $m$.
\end{prop}

\begin{prop}
If $[C_1 \ \ C_2] = [C_3 \ \ C_4] = [0\ \ 0]$, then $[B_1 \ \ B_2] = [B_3 \ \ B_4] = [0\ \ 0]$ as well, and $(\scM, \I)$ and $(\Mbar, \Ibar)$ are each contact equivalent to the Monge-Amp\`ere system representing the standard wave equation.
\end{prop}

\begin{prop}\label{one-C-vector-zero}
If $[C_1 \ \ C_2] = [0\ \ 0]$ (resp., $[C_3 \ \ C_4] = [0\ \ 0]$), then $[B_1 \ \ B_2] = [0\ \ 0]$ (resp., $[B_3 \ \ B_4] = [0\ \ 0]$) as well, and $(\scM, \I)$ and $(\Mbar, \Ibar)$ are each Monge-integrable.
\end{prop}


\begin{prop}
Suppose that the vectors $[C_1 \ \ C_2]$ and $[C_3 \ \ C_4]$ are both nonzero.  Then the vectors $[B_1 \ \ B_2]$ and $[B_3 \ \ B_4]$ are either both zero or both nonzero.
\end{prop}

If $[B_1 \ \ B_2] = [B_3 \ \ B_4] = [0\ \ 0]$, then the differential ideal generated by $\{\om^1, \om^2, \om^3, \om^4\}$ is a Frobenius system.  (The converse is true as well.) It follows that locally, there exists a 4-manifold $\scrV$ which is a quotient of
$\B$ and for which the 1-forms $\om^1, \om^2, \om^3, \om^4$ are
semi-basic for the projection $\rho:\B \to \scrV$.  (Here ``locally'' refers
to the fact that any point in $\B$ has a neighborhood which
possesses such
a quotient, and ``semi-basic'' means that the
restrictions of the $\om^i$ to the fibers of the projection vanish
identically.  See \cite{CFB} for details.)  In fact, this quotient
factors through each of the quotients $\pi: \B \to \scM$ and $\pibar: \B \to \Mbar$, as shown
by the following commutative diagram.
\setlength{\unitlength}{2pt}
\begin{center}
\begin{picture}(40,40)(0,10)
\put(5,28){\makebox(0,0){$\scM$}}
\put(35,28){\makebox(0,0){$\Mbar$}}
\put(20,45){\makebox(0,0){$\B$}}
\put(16,41){\vector(-1,-1){9}}
\put(24,41){\vector(1,-1){9}}
\put(7,37){$\pi$}
\put(31,37){$\pibar$}
%\put(7,38){\makebox(0,0){$\scriptstyle{\pi}$}}
%\put(33,38){\makebox(0,0){$\scriptstyle{\pibar}$}}
\put(20,40){\vector(0,-1){23}}
\put(23,27){\makebox(0,0){$\rho$}}
\put(20,13){\makebox(0,0){$\scrV$}}
\put(7,25){\vector(1,-1){9}}
\put(33,25){\vector(-1,-1){9}}
\end{picture}
\end{center}
The vanishing of the vector $[B_1\ \ B_2]$ implies that the
span of $\{\w^1, \w^2\}$ is unchanged along the fibers of $\rho$,
and is thus the pullback via $\rho$ of a well-defined rank 2 sub-bundle of $T^*\scrV$.
When the vector $[B_3 \ \ B_4]$ also vanishes,
the ideal $\{\om^1 \& \om^2,\  \om^3 \& \om^4\}$
is the pullback via $\rho$ of a well-defined hyperbolic system
$\mathcal{H}$ of class 0 on $\scrV$,
and $\I, \Ibar$ are both integrable extensions of $\mathcal{H}$.

\BT/s of this type are called {\em holonomic}.  One can test whether a \BT/ is holonomic by checking whether the Pfaffian system on $\B$ spanned by the intersection of the basic forms for $\pi$ with the basic forms for $\pibar$ is Frobenius.  Note that
the basic  forms for $\pi$ are spanned by the {\em Cartan system}\footnote{The {\em Cartan system} for a given EDS $\I$ is the smallest Frobenius system that contains $\I$.} of $\I$, and the basic forms for $\pibar$ are spanned by the Cartan system of $\Ibar$.


Holonomic \BT/s are generally considered less interesting than non-holonomic \BT/s because of their limited capacity to generate new solutions,
which we now explain. Given an integral surface $\scN$ of $(\scM, \I)$, solving the Frobenius system $\J$ on $\pi^{-1}(\scN)$ produces a 1-parameter family of integral surfaces $\overline{\scN}_{\lambda}$ of $(\Mbar, \Ibar)$.  Reversing the process, beginning with any one of the integral manifolds $\overline{\scN}_{\lambda}$, in turn produces a 1-parameter family of integral surfaces $\scN_{\lambda, \mu}$ of $(\scM, \I)$.  In general, this results in a 2-parameter family of integral surfaces of $(\scM, \I)$, and iterating the process produces an ever-increasing family of new integral surfaces for both Monge-Amp\`ere systems.  %(Indeed, iterating this process for the sine-Gordon system \eqref{SGsingleBT}, beginning with the vacuum solution $u(x,y)=0$, produces the $k$-soliton solutions after $k$ iterations.)

For example, consider the system \eqref{Liouvillewave}.  If we substitute the trivial solution $z(x,y)=0$ of the wave equation into \eqref{Liouvillewave}, the resulting overdetermined PDE system for $u$ yields the 1-parameter family of solutions
\begin{equation}
u(x,y) = -2\ln (-x -\tfrac{1}{2} y - c_1) \label{Liouville-u1}
\end{equation}
to Liouville's equation.  Reversing the process, substituting \eqref{Liouville-u1} into \eqref{Liouvillewave} produces a PDE system for $z$ which has a 2-parameter family of solutions:
\begin{equation}
z(x,y) = 2 \ln (-y - c_2) - 2 \ln (2x + 2c_1 - c_2) . \label{Liouville-z2}
\end{equation}
Finally, substituting \eqref{Liouville-z2} into \eqref{Liouvillewave} produces a PDE system for $u$ which has a 3-parameter family of solutions:
\begin{equation}
u(x,y) = -2\ln \big{(}c_3 xy  + (c_2 c_3 - 1) x +
(c_1 c_1 - \tfrac{1}{2} c_2 c_3 -\tfrac{1}{2}) y +
(c_1 c_2 c_3 - \tfrac{1}{2} c_2^2 c_3 -  c_1) \big{)}. \label{Liouville-u3}
\end{equation}
It is clear that the solutions \eqref{Liouville-u1} form a proper subset of the solutions \eqref{Liouville-u3}, since the argument of the latter contains an $xy$ term.


For the system \eqref{Liouvillewave}, and for non-holonomic \BT/s in general, successive iterations of this process continue to produce new solutions. However, if the \BT/ is holonomic, then all integral surfaces of $(\scM, \I)$ and $(\Mbar, \Ibar)$ produced by this process must lie in the inverse image of a single integral surface of $(\scrV, \mathcal{H})$.  It follows that successive iterations can produce no more than a 1-parameter family of integral surfaces for each Monge-Amp\`ere system.






