%\setcounter{section}{3}%so that this will be section 4
\section{Proof that Darboux implies B\"acklund}
\label{wprovebacklund}

\subsection{The non-Monge-integrable case}\label{NMIcase}
In this subsection $(\scM, \I)$ is assumed to be a hyperbolic \MA/ system which is
Darboux-integrable after one prolongation, but not Monge-integrable.
We will construct a canonical coframing associated to the prolongation.  We will then use this coframing to construct an integrable extension $\J$ of $(\scM, \I)$ in such a way that $\J$ defines a B\"acklund transformation between $(\scM, \I)$ and the standard wave equation $Z_{XY}=0$.

\begin{lemma}\label{NMIm5coframe} Near any point of $\scM$, there exists a coframing $(\theta,\pi_1,\pi_2,\eta^1,\eta^2)$
such that $\theta$ spans the 1-forms of $\I$, and the characteristic systems
$\calC_1, \calC_2$ of $\I$ have
derived flags
$$
\calC_1 = \{ \theta,\pi_1,\eta^1\} \supset \{\pi_1, \eta^1\} \supset \{\eta^1 \} =\calC_1^{(\infty)},
\qquad
\calC_2 = \{ \theta,\pi_2,\eta^2\} \supset \{\pi_2, \eta^2\} \supset \{\eta^2 \}=\calC_2^{(\infty)}.
$$
\end{lemma}
\begin{proof}
By a result of Jur\'a\v{s} \cite{Juras}, $(\scM,\I)$ is locally contact equivalent
to a system encoding a PDE of the form
\[ u_{xy}=F(x,y,u,p,q). \]
Thus, there are local coordinates
$x,y,u,p,q$ near the given point of $\scM$ such that $\I$ is generated by the 1-form
$\theta= du - p\, dx- q\, dy$ and
the 2-forms $(dp - F\,dy) \& dx$ and $(dq - F\,dx) \& dy$.  It is easy to verify that
the coframing given by $\theta$, $\eta^1 = dx$, $\eta^2 = dy$, $\pi_1 = dp - F\, dy - F_q \theta$,
and $\pi_2 = dq - F \,dx - F_p \theta$ has the properties claimed.
\end{proof}

In terms of the local coframing on $\scM$ given by the lemma, the
prolongation $(\scM', \I')$ is defined as follows:
let $\scM' = \scM \times \R^2$, with coordinates $r,t$ on the $\R^2$ factor,
and let $\I'$ be the Pfaffian system on $\scM'$
generated by $\theta$ and the forms
\begin{equation}\label{theta12def}
\theta_1 = \pi_1 - r \eta^1, \qquad \theta_2 = \pi_2 - t \eta^2.
\end{equation}

\begin{lemma}\label{n7coframe} Near any point of $\scM'$ there exists a coframing
$(\theta,\theta_1,\theta_2,\eta^1,\eta^2,\pi_3,\pi_4)$ such that
$\I'$ is generated by $\theta,\theta_1,\theta_2$, satisfying
\begin{equation}\label{nmi-Darboux-structure-0}
\begin{aligned}
d\theta & = -\theta_1 \& \eta^1 - \theta_2 \& \eta^2 \mod{\theta} %\notag
\\
d\theta_1 & = -\pi_3 \& \eta^1 \mod{\theta, \theta_1} %\label{nmi-Darboux-structure-0}
\\
d\theta_2 & = -\pi_4 \& \eta^2 \mod{\theta, \theta_2},
%\notag
\end{aligned}
\end{equation}
with the derived flags of the characteristic systems of $\I'$ given by
\begin{gather*}
\Cprime1  = \{ \theta, \theta_1, \theta_2, \eta^1, \pi_3 \}
 \supset  \{ \theta, \theta_1, \eta^1, \pi_3 \}  \supset
 \{ \theta_1, \eta^1, \pi_3 \} \supset  \{ \eta^1, \pi_3 \}  = \Cprime1^{(\infty)},
\\
\Cprime2  = \{\theta, \theta_1, \theta_2, \eta^2, \pi_4\}
\supset \{ \theta, \theta_2, \eta^2, \pi_4 \} \supset
\{\theta_2, \eta^2, \pi_4\} \supset  \{\eta^2, \pi_4\} = \Cprime2^{(\infty)}.
\end{gather*}
\end{lemma}

\begin{proof}Let $\theta,\eta^1=dx, \eta^2=dy$ be part of the local coframing
on $\scM$ (pulled back to $\scM')$ constructed in the proof of Lemma \ref{NMIm5coframe},
and let $\theta_1,\theta_2$ be defined as in \eqref{theta12def}. Then
\begin{align*}
d\theta_1 &\equiv -(dr - (D_x F) dy) \& dx \quad \mod{\theta,\theta_1}, \\
d\theta_2 &\equiv -(dt - (D_y F) dx) \& dy \quad \mod{\theta,\theta_2},
\end{align*}
where
\begin{align*}
D_x F &= F_x + F_u p + F_p r + F_q F,\\
D_y F &= F_y + F_u q + F_p F + F_q t.
\end{align*}
For the moment, let $\pi_3 = dr - (D_x F)dy$.
Because $d\pi_3  \equiv 0$ modulo $dx, \pi_3, \theta,\theta_1,\theta_2$,
it follows that $\pi_3$ lies in $\Cprime1^{(1)}$.  Moreover, we may subtract a multiple of
$\theta$ from $\pi_3$ to ensure that $\pi_3$ lies in $\Cprime1^{(2)}$.

Next, we prove that the last derived system of $\Cprime1$ has rank 2, rather than rank 3.
(A similar argument applies to $\Cprime2$.)  Suppose that $\Cprime1^{(2)} = \{\theta_1, \eta^1, \pi_3\}$ is integrable.  From \eqref{theta12def}, it is clear that this is equivalent to the statement that $\{\pi_1, \eta^1, \pi_3\}$ is integrable---i.e., that $d\pi_1 \equiv d\eta^1 \equiv d\pi_3 \equiv 0$ modulo $\pi_1, \eta^1, \pi_3$.  But $\pi_1$ and $\eta^1$ are both well-defined on $\scM$, so their exterior derivatives do not involve $\pi_3$.  It follows that $d\pi_1 \equiv d\eta^1 \equiv 0$ modulo $\pi_1, \eta^1$, and $\calC_1^{(1)} = \{\pi_1, \eta^1\}$ is integrable, contrary to the hypothesis that $(\scM, \I)$ is not Monge-integrable.



%It follows from \eqref{nmi-Darboux-structure-0} that there are functions $A,B,C,D,E$ such that
%$$d\theta_1 \equiv - \pi_3 \& \eta^1 + \theta \& (A\theta_2 + B\eta^2 + C\eta^1 + D\pi_1 + E\pi_2)
%\mod{\theta_1}.$$
%However, computing $0 = d(d\theta) \equiv d\theta_1 \& \eta^1$ mod $\theta,\theta_1,\theta_2,\eta^2$
%shows that $D=E=0$; moreover, we can add a multiple of $\eta^1$ to $\pi_3$ to make $C=0$.
%If $\theta_1$ lies in the last derived system of $\Cprime1$, then $A$ and $B$ must be zero,
%and it follows that $\{ \theta_1, \eta^1\}$ is Frobenius as well.  But the span of
%these 1-forms is the same as the pullback from $\scM$ of the system $\{\pi_1,\eta^1\}$.
%Since we have assumed that the \MA/ $\I$ is not Monge-integrable, then
%the last derived system of $\Cprime1$ cannot have rank 3.
%By our assumption of Darboux-integrability, the derived flag terminates in a
%rank 2 Frobenius system, and we may adjust $\pi_3$ to lie in that system.
%A similar argument gives the corresponding properties for $\Cprime2$ and $\pi_4$.

\end{proof}

The conditions in Lemma \ref{n7coframe} are preserved by changes of coframing
of the form
\begin{equation} \label{Darboux-freedom}
\begin{bmatrix}
\tilde{\theta}\\[0.1in] \tilde{\theta}_1 \\[0.1in] \tilde{\theta}_2 \\[0.1in] \tilde{\eta}^1 \\[0.1in] \tilde{\eta}^2 \\[0.1in] \tilde{\pi}_3 \\[0.1in] \tilde{\pi}_4
\end{bmatrix} =
\begin{bmatrix}
c & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & a_1 c & 0 & 0 & 0 & 0 & 0  \\[0.1in]
0 & 0 & a_2 c & 0 & 0 & 0 & 0  \\[0.1in]
0 & 0 & 0 & a_1^{-1} & 0 & 0 & 0  \\[0.1in]
0 & 0 & 0 & 0 & a_2^{-1} & 0 & 0   \\[0.1in]
0 & 0 & 0 & b_1 & 0 & a_1^2 c & 0  \\[0.1in]
0 & 0 & 0 & 0 & b_2 & 0 & a_2^2 c
\end{bmatrix}^{-1}
\begin{bmatrix}
\theta\\[0.1in] \theta_1 \\[0.1in] \theta_2 \\[0.1in] \eta^1 \\[0.1in] \eta^2 \\[0.1in] \pi_3 \\[0.1in] \pi_4
\end{bmatrix},
\end{equation}
with $a_1, a_2, c \neq 0$.  Let $G\subset GL(7,\R)$ be the group
of such transformations, and let $\calP$ be the $G$-structure on $\scM'$
of which the coframing of Lemma \ref{n7coframe} is a section.

After absorbing as much torsion as possible and differentiating to uncover relations among the torsion,
$\calP$ has structure equations
\begin{multline}\hskip -.2in
\begin{bmatrix} d\theta\\[0.1in] d\theta_1\\[0.1in] d\theta_2 \\[0.1in] d\eta^1\\[0.1in] d\eta^2 \\[0.1in] d\pi_3 \\[0.1in] d\pi_4
\end{bmatrix} =-
%\begin{bmatrix}
{\setlength\arraycolsep{0.5pt}
\left[  \begin{array}{c c c c c c c}
\gamma & 0 & 0 & 0 & 0 & 0 & 0  \\[0.1in]
0 & \ \gamma + \alp_1 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & 0 & \gamma + \alp_2 & 0 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & -\alp_1 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & 0 & -\alp_2 & 0 & 0 \\[0.1in]
0 & 0 & 0 & \bet_1 & 0 & \gamma + 2\alp_1 & 0 \\[0.1in]
0 & 0 & 0 & 0 & \bet_2 & 0 & \gamma + 2\alp_2
\end{array}\right]
}
% \end{bmatrix}
 \&
\begin{bmatrix} \theta\\[0.1in] \theta_1\\[0.1in] \theta_2 \\[0.1in] \eta^1\\[0.1in] \eta^2 \\[0.1in] \pi_3 \\[0.1in] \pi_4
\end{bmatrix}
%\\[0.2in]
%\label{Darboux-structure}
- \begin{bmatrix}
\theta_1 \& \eta^1 \! + \! \theta_2 \& \eta^2 \\[0.1in]
\pi_3 \& \eta^1 \! + \! (A_2 \theta_2 \! + \! B_2 \eta^2) \& \theta \\[0.1in]
\pi_4 \& \eta^2 \! + \! (A_1 \theta_1 \! + \! B_1 \eta^1) \& \theta \\[0.1in]
0 \\[0.1in]
0 \\[0.1in]
2C_1 \theta_1 \& \pi_3   \\[0.1in]
2C_2 \theta_2 \& \pi_4
\end{bmatrix}\hskip -2pt
.
\end{multline}

Because of the dimensions of the derived flags of the characteristic systems (given
in Lemma \ref{n7coframe}),  $A_1, B_1$ are not both zero, and $A_2, B_2$ are not both zero.
%Without loss of generality, we may assume that $(\scM, \I)$ represents a wave equation of the form
%\[ u_{xy} = F(x,y,u,p,q). \]
Furthermore, we can choose a local section $\sigma: \scM' \to \calP$ satisfying the conditions that $\eta^1 = dx$, $\eta^2 = dy$, and the forms $\pi_3, \pi_4$ are integrable; i.e.,
\[ d\pi_3 \equiv 0 \mod{\pi_3}, \qquad d\pi_4 \equiv 0 \mod{\pi_4}. \]
To see why, note that $\{ \eta^1, \pi_3 \}$ is a Frobenius system, and so it is spanned locally by two exact 1-forms.  Thus we can adjust $\pi_3$ by adding multiples of $\eta^1$ in order to make it a multiple of an exact form.   Similarly, we can add multiples of $\eta^2$ to  $\pi_4$ in order to make $\pi_4$ a multiple of an exact form.  However, we cannot independently scale $\pi_3$ and $\pi_4$ to make both of them exact.

This choice of section is not unique; it is determined up to a transformation of the form
\begin{equation}
\begin{bmatrix}
\tilde{\theta}\\[0.1in] \tilde{\theta}_1 \\[0.1in] \tilde{\theta}_2 \\[0.1in] \tilde{\eta}^1 \\[0.1in] \tilde{\eta}^2 \\[0.1in] \tilde{\pi}_3 \\[0.1in] \tilde{\pi}_4
\end{bmatrix} =
\begin{bmatrix}
c & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & c & 0 & 0 & 0 & 0 & 0  \\[0.1in]
0 & 0 & c & 0 & 0 & 0 & 0  \\[0.1in]
0 & 0 & 0 & 1 & 0 & 0 & 0  \\[0.1in]
0 & 0 & 0 & 0 & 1 & 0 & 0   \\[0.1in]
0 & 0 & 0 & 0 & 0 &  c & 0  \\[0.1in]
0 & 0 & 0 & 0 & 0 & 0 &  c
\end{bmatrix}^{-1}
\begin{bmatrix}
\theta\\[0.1in] \theta_1 \\[0.1in] \theta_2 \\[0.1in] \eta^1 \\[0.1in] \eta^2 \\[0.1in] \pi_3 \\[0.1in] \pi_4
\end{bmatrix}
\end{equation}
with $c \neq 0$.  However, we can make the choice of $\sigma$ unique (albeit slightly non-canonical) as follows: since $\pi_3$, $\pi_4$ are integrable 1-forms, we must have
\[ \pi_3 = e^g d\xi_1, \qquad \pi_4 = e^h d\xi_2 \]
for some functions $\xi_1, \xi_2, f, g$ on $\scM'$.  Using the remaining scaling freedom, we can arrange that $h=-g$; the resulting coframing $\sigma: \calP \to \scM$ is uniquely determined.


When we pull back the structure equations via $\sigma$, the pseudoconnection forms $\alp_1$, $\alp_2$, $\beta_1$, $\beta_2$, $\gamma$ become semi-basic.  By making use of the remaining ambiguity in these forms and the conditions imposed thus far on the coframing, we can assume that
\begin{align*}
\alp_1 & = (D_1 + E_1)\eta^1 \\
\alp_2 & = (D_2 + E_2)\eta^2 \\
\beta_1 & = 2 D_1 \pi_3 \\
\beta_2 & = 2 D_2 \pi_4 \\
\gamma & = -C_1 \theta_1 - C_2 \theta_2 - E_1 \eta^1 - E_2 \eta^2 + F_1 \pi_3 + F_2 \pi_4
\end{align*}
for some functions $C_i, D_i, E_i, F_i$.
Then the structure equations for this coframing become:
\begin{align}
d\theta & = \theta \& (-C_1 \theta_1 - C_2 \theta_2 - E_1 \eta^1 - E_2 \eta^2 + F_1 \pi_3 + F_2 \pi_4)  - \theta_1 \& \eta^1 - \theta_2 \& \eta^2   \notag \\
d\theta_1 & = \theta_1 \& (-C_2 \theta_2 - E_2 \eta^2 + F_1 \pi_3 + F_2 \pi_4)  + D_1 \theta_1 \& \eta^1  - \pi_3 \& \eta^1 + \theta \& (A_2 \theta_2 + B_2 \eta^2)   \notag \\
d\theta_2 & = \theta_2 \& (-C_1 \theta_1 - E_1 \eta^1 + F_1 \pi_3 + F_2 \pi_4)  + D_2 \theta_2 \& \eta^2  - \pi_4 \& \eta^2 + \theta \& (A_1 \theta_1 + B_1 \eta^1) \notag \\
d\eta^1 & = 0   \label{Darboux-section-structure} \\
d\eta^2 & =  0  \notag \\
d\pi_3 & = \pi_3 \& (C_1 \theta_1 - C_2 \theta_2 + E_1 \eta^1 - E_2 \eta^2 + F_2 \pi_4)    \notag \\
d\pi_4 & = \pi_4 \& (-C_1 \theta_1 + C_2 \theta_2 - E_1 \eta^1 + E_2 \eta^2 + F_1 \pi_3).   \notag
\end{align}
(Note that these torsion functions are completely unrelated to those in \S \ref{wsetup} and \S \ref{wprovedarboux}.)

As in \S \ref{wprovedarboux}, we will need to compute relations among the derivatives of the torsion functions in order to show that $(\scM, \I)$ has a B\"acklund transformation to the wave equation.  We begin by differentiating the structure equations \eqref{Darboux-section-structure}.  Using notation similar to that in \S \ref{wprovedarboux}, we denote derivatives as, e.g.,
\[ dA_1 = A_{1,0} \theta + A_{1,1} \theta_1 + A_{1,2} \theta_2 + A_{1,3} \eta^1 + A_{1,4} \eta^2 + A_{1,5} \pi_3 + A_{1,6} \pi_4. \]
(Note that since this coframing is defined on a different manifold from that in \S \ref{wprovedarboux}, the indexing of the derivatives is different as well.)

Computing $d(d\theta) = d(d\theta_1) = d(d\theta_2) = d(d\pi_3)  = d(d\pi_4) = 0$ yields the following equations for the derivatives of the torsion functions:
\begin{align}
dA_1 & = A_{1,0} \theta + A_{1,1} \theta_1 -3 A_1 C_2 \theta_2 + A_{1,3} \eta^1 - A_1(D_2 + 2 E_2) \eta^2 + A_1 F_3 \pi_3 + A_1 F_2 \pi_4 \notag \\
dA_2 & = A_{2,0} \theta - 3 A_2 C_1 \theta_1 + A_{2,2} \theta_2 - A_2(D_1 + 2 E_1) \eta^1 + A_{2,4} \eta^2 + A_2 F_3 \pi_3 + A_2 F_2 \pi_4 \notag \\
dB_1 & = B_{1,0} \theta + (A_{1,3} - A_1 D_1) \theta_1 - 2 B_1 C_2 \theta_2 + B_{1,3} \eta^1 - B_1(D_2 + E_2) \eta^2 + A_1 \pi_3 \notag \\
dB_2 & = B_{2,0} \theta - 2 B_2 C_1 \theta_1 + (A_{2,4} - A_2 D_2) \theta_2  - B_2(D_1 + E_1) \eta^1 + B_{2,4} \eta^2 + A_2 \pi_4 \notag \\
dC_1 & = A_1 C_2 \theta + C_{1,1} \theta_1 - C_1 C_2 \theta_2 + C_{1,3} \eta^1 - (\tfrac{1}{2} A_1 + C_1 E_2) \eta^2 + C_{1,5} \pi_3 + C_1 F_2 \pi_4 \notag \\
dC_2 & = A_2 C_1 \theta - C_1 C_2 \theta_1 + C_{2,2} \theta_2 - (\tfrac{1}{2} A_2 + C_2 E_1) \eta^1 + C_{2,4} \eta^2 + C_2 F_1 \pi_3 + C_{2,6} \pi_4 \label{dtorsion} \\
dD_1 & = B_1 C_2 \theta + D_{1,1} \theta_1 + \tfrac{3}{2} A_2 \theta_2 + D_{1,3} \eta^1 + \tfrac{1}{2}(3 B_2 - B_1) \eta^2 + (2 C_1 - E_{1,5}) \pi_3 \notag \\
dD_2 & = B_2 C_1 \theta + \tfrac{3}{2} A_1 \theta_1 + D_{2,2} \theta_2 + \tfrac{1}{2}(3 B_1 - B_2) \eta^1 + D_{2,4} \eta^2 + (2 C_2 - E_{2,6}) \pi_4 \notag \\
dE_1 & = B_1 C_2 \theta + (C_{1,3} - C_1 D_1) \theta_1 + \tfrac{1}{2} A_2 \theta_2 + E_{1,3} \eta^1 + \tfrac{1}{2}(B_2 - B_1) \eta^2 + E_{1,5} \pi_3 \notag \\
dE_2 & = B_2 C_1 \theta + \tfrac{1}{2} A_1 \theta_1 + (C_{2,4} - C_2 D_2) \theta_2 + \tfrac{1}{2}(B_1 - B_2) \eta^1 + E_{2,4} \eta^2 + E_{2,6} \pi_4 \notag \\
dF_1 & = (2 C_1 F_1 - C_{1,5}) \theta_1 - C_2 F_1 \theta_2 + (C_1 + E_1 F_1 - E_{1,5}) \eta^1 - E_2 F_1 \eta^2 + F_{1,5} \pi_3 + F_{1,6} \pi_4\notag \\
dF_2 & = -C_1 F_2 \theta_1 + (2 C_2 F_2 - C_{2,6}) \theta_2 - E_1 F_2 \eta^1 + (C_2 + E_2 F_2 - E_{2,6}) \eta^2 + F_{1,6} \pi_3 + F_{2,6} \pi_4. \notag
\end{align}

Because $A_1$ appears as a derivative of $B_1$, and $A_1, B_1$ cannot vanish simultaneously, $B_1$ cannot vanish on any open set in $\scM'$.  In fact, $B_1$ cannot vanish identically on any fiber of the projection $\scM' \to \scM$, and the same is true of $B_2$.   Henceforth we restrict to the dense open set in $\scM'$ where $B_1, B_2$ are both nonzero, and note that this set surjects onto $\scM$.

We may obtain further relations among the derivatives of the torsion functions by differentiating equations \eqref{dtorsion}.  Computing $d(dA_1) \equiv d(dB_1) \equiv 0$ modulo $\theta, \theta_1, \eta^1, \pi_4$ yields
\[
A_{1,0}  = A_1(C_{2,4} - D_{2,2} - C_2 D_2), \qquad
B_{1,0}  = B_1(C_{2,4} - D_{2,2} - C_2 D_2).
\]
Then computing $d(dB_1) \equiv 0$ modulo $\theta, \theta_1, \eta^1$ yields
$ C_{2,6} = C_2 F_2. $
Similar considerations of $d(dA_2)$ and $d(dB_2)$ show that
\begin{gather*}
A_{2,0}  = A_2(C_{1,3} - D_{1,1} - C_1 D_1) , \qquad
B_{2,0}  = B_2(C_{1,3} - D_{1,1} - C_1 D_1), \qquad
C_{1,5} = C_1 F_1.
\end{gather*}

It will now be convenient to derive several equations and solve them simultaneously.  First, $d(dC_1) \equiv 0$ modulo $\theta_1, \eta^1$ implies that
\begin{gather}
A_1(C_{2,2} - C_2^2) = A_2(C_{1,1} - C_1^2) \label{eqn1forH} \\
A_1(3C_{2,4} - D_{2,2} - 3 C_2 D_2) = 2 B_2 (C_{1,1} - C_1^2).
\end{gather}
Additionally, $d(dC_2) \equiv 0$ modulo $\theta_2, \eta^2$ implies that
\begin{equation}
A_2(3C_{1,3} - D_{1,1} - 3 C_1 D_1) = 2 B_1 (C_{2,2} - C_2^2).
\end{equation}
Finally, $d(dD_1) \equiv 0$ modulo $\theta_1, \eta^1, \pi_3$ implies that
\begin{equation}
B_1(3C_{2,4} - D_{2,2} - 3 C_2 D_2) = B_2(3C_{1,3} - D_{1,1} - 3 C_1 D_1). \label{eqn4forH}
\end{equation}
The general solution to equations \eqref{eqn1forH}-\eqref{eqn4forH} is most easily expressed in terms of a new torsion function $H$, such that
\begin{align*}
C_{1,1} &= C_1^2 - A_1 H, &
D_{1,1} &= 3(C_{1,3} - C_1 D_1) + 2 B_1 H, \\
C_{2,2} &= C_2^2 - A_2 H, &
D_{2,2} &= 3(C_{2,4} - C_2 D_2) + 2 B_2 H.
\end{align*}

Next, we need information about the derivatives of $C_{1,3}$ and $C_{2,4}$.  Computing $d(dC_1) \equiv 0$ modulo $\theta_1$ yields
\begin{align*}
dC_{1,3} & = (C_2 A_{1,3} - \tfrac{1}{2} A_1 A_2 + B_1 C_1 C_2) \theta + C_{1,31} \theta_1 + (\tfrac{1}{2} A_2 C_1 - C_2 C_{1,3}) \theta_2 \\
& \qquad  + C_{1,33} \eta^1  - \tfrac{1}{2}(A_{1,3} + 2 E_2 C_{1,3} + C_1(B_1 - B_2)) \eta^2 \\
& \qquad \qquad + (F_1 C_{1,3} - C_1 E_{1,5} + 2 C_1^2 - A_1 H) \pi_3  + F_2 C_{1,3} \pi_4,
\end{align*}
and computing $d(dC_2) \equiv 0$ modulo $\theta_2$ yields
\begin{align*}
dC_{2,4} & = (C_1 A_{2,4} - \tfrac{1}{2} A_1 A_2 + B_2 C_1 C_2) \theta + (\tfrac{1}{2} A_1 C_2 - C_1 C_{2,4}) \theta_1 + C_{2,42} \theta_2 \\
& \qquad  - \tfrac{1}{2}(A_{2,4} + 2 E_1 C_{2,4} + C_2(B_2 - B_1)) \eta^1
 + C_{2,44} \eta^2 \\
 & \qquad \qquad  + F_1 C_{2,4} \pi_3 + (F_2 C_{2,4} - C_2 E_{2,4} + 2 C_2^2 - A_2 H) \pi_4 .
\end{align*}

Now, computing $d(dC_1) = d(dC_2) = 0$, $d(dD_1) \equiv 0$ modulo $\eta^1, \pi_3$, and $d(dD_2) \equiv 0$ modulo $\eta^2, \pi_4$ yields four different expressions for $dH$.  Taking linear combinations of these expressions shows that
\begin{gather}
A_1 (A_{1,3} - A_1 D_1) = B_1 (A_{1,1} - A_1 C_1) \label{J1eq} \\
A_2 (A_{2,4} - A_2 D_2) = B_2 (A_{2,2} - A_2 C_2) \label{J2eq}.
\end{gather}
Equations \eqref{J1eq} and \eqref{J2eq} may be solved by introducing new torsion functions $J_1, J_2$,
such that
\begin{align*}
A_{1,1} & = A_1 C_1 + A_1 J_1 &
A_{2,2} & = A_2 C_2 + A_2 J_2 \\
A_{1,3} & = A_1 D_1 + B_1 J_1 &
A_{2,4} & = A_2 D_2 + B_2 J_2.
\end{align*}
Then the various expressions for $dH$ may be combined to show that
\begin{gather*}
C_{1,31} = 2 C_1 C_{1,3} - C_1^2 D_1 + A_1 C_2 - \tfrac{1}{2} A_1 J_2 - (A_1 D_1 + B_1 J_1) H \\
C_{2,42} = 2 C_2 C_{2,4} - C_2^2 D_2 + A_2 C_1 - \tfrac{1}{2} A_2 J_1 - (A_2 D_2 + B_2 J_2) H,
\end{gather*}
and
\begin{multline*}
dH = \left(H(C_{1,3} - C_1 D_1 + B_1 H + C_{2,4} - C_2 D_2 + B_2 H) - \tfrac{1}{2} C_1 J_2 + C_2 J_1\right) \theta \\ + C_1 H \theta_1 + C_2 H \theta_2 + (D_1 H + \tfrac{1}{2} J_2) \eta^1 + (D_2 H + \tfrac{1}{2} J_1) \eta^2
+ F_1 H \pi_3 + F_2 H \pi_4.
\end{multline*}
They also imply the relation
\begin{equation}
H(C_{1,3} - C_1 D_1 + B_1 H) - \tfrac{1}{2} C_1 J_2 = H(C_{2,4} - C_2 D_2 + B_2 H) - \tfrac{1}{2} C_2 J_1. \label{zero1}
\end{equation}
The equations for $dA_1, dA_2$ now take the form:
\begin{multline}
dA_1 = -2A_1(C_{2,4} - C_2 D_2 + B_2 H)  \theta + A_1(C_1 + J_1) \theta_1 -3 A_1 C_2 \theta_2 \\
+ (A_1 D_1 + B_1 J_1) \eta^1  - A_1(D_2 + 2 E_2) \eta^2 + A_1 F_3 \pi_3 + A_1 F_2 \pi_4 \label{betterdA1eq}
\end{multline}
\begin{multline}
dA_2 =  -2A_2(C_{1,3} - C_1 D_1 + B_1 H) \theta - 3 A_2 C_1 \theta_1 + A_2(C_2 + J_2) \theta_2 \\
 - A_2(D_1 + 2 E_1) \eta^1 + (A_2 D_2 + B_2 J_2) \eta^2 + A_2 F_3 \pi_3 + A_2 F_2 \pi_4. \label{betterdA2eq}
\end{multline}

Now, computing $d(dB_1) \equiv 0$ modulo $\theta_1, \eta^1$ yields
\[ C_{2,44} = (D_2 - E_2)C_{2,4} + C_2 D_{2,4} - H B_{2,4} - (D_2 + E_2) B_2 H + C_2 D_2 E_2 + B_2 (C_1 - J_1). \]
Similarly, computing $d(dB_2) \equiv 0$ modulo $\theta_2, \eta^2$ yields
\[ C_{1,33} = (D_1 - E_1)C_{1,3} + C_1 D_{1,3} - H B_{1,3} - (D_1 + E_1) B_1 H + C_1 D_1 E_1 + B_1 (C_2 - J_2). \]
Next, computing $d(dA_1) = 0$ shows that
\begin{align*}
dJ_1 & = 4 A_1 C_2 \theta + (C_1 J_1 + A_1 K_1) \theta_1 - C_2 J_1 \theta_2 \\
& \qquad + (2C_{1,3} - 2C_1 D_1 + 2B_1 H + 2C_{2,4} - 2C_2 D_2 + 2B_2 H + D_1 J_1 + B_1 K_1) \eta^1 \\
& \qquad \qquad  - (2 A_1 + E_2 J_1) \eta^2 + F_1 J_1 \pi_3 + F_2 J_1 \pi_4
\end{align*}
for some function $K_1$.  Similarly, computing $d(dA_2) = 0$ shows that
\begin{align*}
dJ_2 & = 4 A_2 C_1 \theta - C_1 J_2 \theta_1 + (C_2 J_2 + A_2 K_2) \theta_2 - (2 A_2 + E_1 J_2) \eta^1\\
& \qquad + (2C_{1,3} - 2C_1 D_1 + 2B_1 H + 2C_{2,4} - 2C_2 D_2 + 2B_2 H + D_2 J_2 + B_2 K_2) \eta^2 \\
& \qquad \qquad   + F_1 J_2 \pi_3 + F_2 J_2 \pi_4
\end{align*}
for some function $K_2$.  But now computing $d(dH) \equiv 0$ modulo $\theta, \theta_1, \theta_2$ yields
\[ B_1 (K_1 + 4 H) = B_2 (K_2 + 4 H). \]
It follows that
\begin{align*}
K_1 & = -4H + B_2 M \\
K_2 & = -4H + B_1 M
\end{align*}
for some function $M$.  Computing $d(dH) \equiv 0$ modulo $\theta_2, \eta^1$ and $d(dH) \equiv 0$ modulo $\theta_1, \eta^2$ shows that
\begin{equation}
 A_1 M = A_2 M = 0. \label{Meqn}
\end{equation}

\begin{claim*}
$M = 0$.
\end{claim*}


\begin{pf}
Suppose not. Then by \eqref{Meqn}, $A_1 = A_2 = 0.$  Therefore, equations \eqref{betterdA1eq} and \eqref{betterdA2eq} reduce to
\begin{gather*}
 0 = dA_1 = B_1 J_1 \eta^1 \\
 0 = dA_2 = B_2 J_2 \eta^2.
\end{gather*}
Since $B_1, B_2$ are nonzero, it follows that $J_1 = J_2 = 0$.  Then
\begin{gather*}
0 = dJ_1 = (2C_{1,3} - 2C_1 D_1 - 2B_1 H + 2C_{2,4} - 2C_2 D_2 + 2B_2 H + B_1 B_2 M) \eta^1 \\
0 = dJ_2 = (2C_{1,3} - 2C_1 D_1 + 2B_1 H + 2C_{2,4} - 2C_2 D_2 - 2B_2 H + B_1 B_2 M) \eta^2.
\end{gather*}
Subtracting the two coefficients above yields
\[ 4 (B_2 - B_1) H = 0, \]
so either $B_1 = B_2$ or $H=0$.

First suppose that $B_1= B_2$. Then
\[ 0 = d(B_2 - B_1) \equiv -2B_1 (C_1 \theta_1 - C_2 \theta_2) \mod{\theta, \eta^1, \eta^2}, \]
so $C_1 = C_2 = 0.$  It follows that $C_{1,3} = C_{2,4} = 0$ as well.  But now
\[ 0 = dJ_1 = B_1^2 M, \]
so $M=0$, as desired.

Now suppose that $H=0$.  Computing $d(dC_{1,3}) - C_1 d(dD_1) \equiv 0$ modulo $\theta$ yields
\[ B_1^2 B_2 M = 0, \]
so $M=0$ in this case as well.
\end{pf}

Finally, computing $d(dH) = 0$, keeping the relation \eqref{zero1} in mind, yields two additional relations:
\begin{gather}
2C_2 (2C_{1,3} - 2 C_1 D_1 + 2 B_1 H) + (2C_2 - J_2) (2C_{2,4} - 2 C_2 D_2 + 2 B_2 H) + A_2 (4 C_1 - J_1) = 0 \label{zero2} \\
(2C_1 - J_1)(2C_{1,3} - 2 C_1 D_1 + 2 B_1 H) + 2 C_1 (2C_{2,4} - 2 C_2 D_2 + 2 B_2 H) + A_1 (4 C_2 - J_2) = 0. \label{zero3}
\end{gather}
We now have all the relations that will be needed for the involutivity calculation below.

\bigskip

Now suppose that $\Mbar=\R^5$ carries a \MA/ system $\Ibar$ representing
the wave equation $Z_{XY}=0$, generated algebraically by the contact form
\begin{equation}\label{defthetabar}
\thetabar = dZ - P\, dX - Q\, dY
\end{equation}
and the 2-forms $dP \& dX$ and $dQ \& dY$.
If there were a B\"acklund transformation $\B \subset \scM \times \Mbar$,
then $Z$ would be a local coordinate on the fibers of $\B \to \scM$
and the functions $X, Y, P, Q$ on $\B$ would satisfy the \BB/ condition
\begin{equation}\label{bcond}
\{dP \& dX, dQ \& dY\} \equiv \{\pi_1 \& \eta^1, \pi_2 \& \eta^2\}
\mod \theta,\thetabar
\end{equation}
(see the definition at the beginning of \S2).
%Moreover, the characteristic systems of $(\Mbar, \Ibar)$ would be $\{\thetabar, dX, dP\}$ and $\{\thetabar, dY, dQ\}$.

Accordingly, we let $\B = \scM \times \R$, with coordinate $Z$ on the second factor.
We will show that, on an open neighborhood of any point of $\B$,
 there exist functions $X,Y,P,Q$ such that the ideal
$\J = \I \cup \{\thetabar\}$ on $\B$ (where $\thetabar$ is defined as in \eqref{defthetabar})
gives a \BT/ between $(\scM, \I)$ and $(\Mbar, \Ibar)$.
We will do this by setting up an EDS whose integral manifolds correspond to functions satisfying these conditions;
once we know that this EDS is involutive, an application of the Cartan-K\"ahler Theorem
 will prove the existence of the desired \BT/s.

%First, the pair $(\B, \J)$ must define an integrable extension of $(\scM, \I)$.
%But we may also think of it as defining an integrable extension of $(\scM', \I')$
%in the obvious way.
Let $\B' = \scM' \times \R$, again with $Z$ as the coordinate on the second factor;
we extend the projection $\scM' \to \scM$ to a projection $\B' \to \B$ by the identity
on the second factor.  It will be convenient to set up our EDS in terms of the
coframing $(\theta,\thetabar,\theta_1,\theta_2,\eta^1,\eta^2,\pi_3,\pi_4)$ on $\B'$.
Thus, we will regard $X, Y, P, Q$ as functions on $\B'$,
but require that
\[ dX, dY, dP, dQ \in \{\theta, \thetabar, \theta_1, \theta_2, \eta^1, \eta^2\} \]
so that they are in fact well-defined on $\B$.
%Second, the condition that $(\Mbar, \Ibar)$ is contact equivalent
%to the wave equation means that $X, P$ and $Y, Q$ are first-order characteristic invariants for $(\Mbar, \Ibar)$.  This is equivalent to the statement that
In order to satisfy the \BB/ condition \eqref{bcond}, we will furthermore require that
\begin{equation}
 \{dX, dP\} \subset \{\theta, \thetabar, \theta_1, \eta^1\}, \qquad \{dY, dQ\} \subset \{\theta, \thetabar, \theta_2, \eta^2\}.  \label{inclusion1}
\end{equation}
From this, and the structure equations \eqref{Darboux-section-structure},
it follows that $\{dX, dP\}$ (resp., $\{dY, dQ\}$) is the largest integrable subsystem of
$\{\theta, \thetabar, \theta_1, \eta^1\}$ (resp., $\{\theta, \thetabar, \theta_2, \eta^2\}$).  Therefore,
\[ \eta^1 = dx \in \{dX, dP\}, \qquad \eta^2 = dy \in \{dY, dQ\}, \]
and by a contact transformation on $\Mbar$, we may assume that $X=x,\ Y=y$.  Thus, we will set
\[  \thetabar = dZ - P\, dx - Q\, dy, \]
and condition \eqref{inclusion1} becomes
\[ dP \in \{\theta, \thetabar, \theta_1, \eta^1\}, \qquad dQ \in \{\theta, \thetabar, \theta_2, \eta^2\}.  \]
Suppose that
\begin{align}
dP & = P_0 \theta + P_{\overline{0}} \thetabar + P_1 \theta_1 + P_3 \eta^1 \label{nmi-dP-eq-1} \\
dQ & = Q_0 \theta + Q_{\overline{0}} \thetabar + Q_2 \theta_2 + Q_4 \eta^2 .\label{nmi-dQ-eq-1}
\end{align}
Observe that normality of the \BT/ requires that $P_1, Q_2 \neq 0$ and $P_1 \neq Q_2$.

\begin{rem}
Equations \eqref{nmi-dP-eq-1}-\eqref{nmi-dQ-eq-1} give an overdetermined system of first-order
partial differential equations for functions $P$ and $Q$.  
The process of generating compatibility conditions
for such systems can be carried out systematically by computing the exterior derivatives of the 1-form
equations, and using the fact that the repeated exterior derivative of a function is zero.
Moreover, applying Cartan's Test for involutivity (see \cite{CFB}, Chapter 7)
to the resulting EDS will tell us
when we can stop differentiating:  if the system is involutive then
no further compatibility conditions arise through differentiation,
and solutions exist that may be constructed by applying
the Cartan-K\"ahler Theorem.
\end{rem}

Differentiating \eqref{nmi-dP-eq-1} modulo $\theta, \thetabar, \theta_1, \eta^1$ yields
\[ (P_0 + P_{\oo} Q_2)\, \theta_2 \& \eta^2 = 0, \]
and differentiating \eqref{nmi-dQ-eq-1} modulo $\theta, \thetabar, \theta_2, \eta^2$ yields
\[ (Q_0 + Q_{\oo} P_1)\, \theta_1 \& \eta^1 = 0. \]
Therefore, because the 1-forms $\theta, \thetabar, \theta_1, \theta_2, \eta^1, \eta^2$ are linearly independent on $\B$, we have $P_0 = -P_{\oo} Q_2,\ Q_0 = -Q_{\oo} P_1$, and we may write
\begin{align}
dP & = P_{\overline{0}} (\thetabar - Q_2 \theta) + P_1 \theta_1 + P_3 \eta^1 \label{nmi-dPQ-eqs1} \\
dQ & = Q_{\overline{0}} (\thetabar - P_1 \theta) + Q_2 \theta_2 + Q_4 \eta^2 .\notag
\end{align}
Note that neither $P_{\oo}$ nor $Q_{\oo}$ can vanish identically: for, if $P_{\oo} = 0$, then differentiating \eqref{nmi-dPQ-eqs1} shows that $P_1 = 0$ as well, which contradicts the hypothesis of normality.  (A similar argument applies to $Q_{\oo}$.)

Differentiating equations \eqref{nmi-dPQ-eqs1} modulo various combinations of 1-forms leads to the following expressions for the exterior derivatives of $P_{\oo}, P_1, P_3, Q_{\oo}, Q_2, Q_4$:
\begin{align}
dP_{\oo} & = -P_{\oo} Q_{\oo 2} \theta + P_{\oo \oo} (\thetabar - Q_2 \theta) + P_{\oo 1} \theta_1 + P_{\oo 3} \eta^1 \notag \\
dP_1 & = P_{\oo 1} (\thetabar - Q_2 \theta) - \frac{(A_1 Q_2 + C_1 Q_{\oo} P_1)}{Q_{\oo}} \theta_1 - C_2 P_1 \theta_2 \notag \\
& \qquad - \frac{(B_1 Q_2 + Q_{\oo}(E_1 P_1 + P_{\oo}(Q_2 - P_1)))}{Q_{\oo}} \eta^1 - E_2 P_1 \eta^2 + F_1 P_1 \pi_3 + F_2 P_1 \pi_4 \notag \\
dP_3 & = (P_{\oo 3} + P_{\oo}^2) (\thetabar - Q_2 \theta) - \frac{(B_1 Q_2 + Q_{\oo}((D_1 + E_1) P_1 + 2P_{\oo}(Q_2 - P_1)))}{Q_{\oo}} \theta_1 \notag \\
& \qquad + P_{33} \eta^1 + P_1 \pi_3 \label{nmi-dPQ-eqs2} \\
dQ_{\oo} & = -Q_{\oo} P_{\oo 1} \theta + Q_{\oo \oo} (\thetabar - P_1 \theta) + Q_{\oo 2} \theta_2 + Q_{\oo 4} \eta^2 \notag \\
dQ_2 & = Q_{\oo 2} (\thetabar - P_1 \theta) - C_1 Q_2 \theta_1 - \frac{(A_2 P_1 + C_2 P_{\oo} Q_2)}{P_{\oo}} \theta_2 \notag \\
& \qquad - E_1 Q_2 \eta^2 - \frac{(B_2 P_1 + P_{\oo}(E_2 Q_2 + Q_{\oo}(P_1 - Q_2)))}{P_{\oo}} \eta^2  + F_1 Q_2 \pi_3 + F_2 Q_2 \pi_4 \notag \\
dQ_4 & = (Q_{\oo 4} + Q_{\oo}^2) (\thetabar - P_1 \theta) - \frac{(B_2 P_1 + P_{\oo}((D_2 + E_2) Q_2 + 2Q_{\oo}(P_1 - Q_2)))}{P_{\oo}} \theta_2 \notag \\
& \qquad + Q_{44} \eta^2 + Q_2 \pi_4. \notag
\end{align}

Now computing $d(dP_1) \equiv 0$ modulo $\thetabar - Q_2 \theta$ and $d(dQ_2) \equiv 0$ modulo $\thetabar - P_1 \theta$ yields
\begin{align}
P_{\oo \oo} & = \frac{1}{Q_{\oo} P_1^2 Q_2 (Q_2 - P_1)} \Big{(}
2 C_1 (P_1 - 2 Q_2)P_1 Q_2 P_{\oo}^2 Q_{\oo} + 2 C_2 P_1^2 Q_2 P_{\oo} Q_{\oo}^2    \notag \\
& \qquad  + 2 (C_{1,3} - C_1 D_1 + B_1 H) P_1^2 Q_2 P_{\oo} Q_{\oo} +  A_1(P_1 - Q_2) Q_2^2 P_{\oo}^2 + A_2 P_1^3 Q_{\oo}^2 \Big{)} \notag \\
Q_{\oo \oo} & =  \frac{1}{P_{\oo} P_1 Q_2^2 (P_1 - Q_2)} \Big{(}
 2 C_1 P_1 Q_2^2 P_{\oo}^2 Q_{\oo}  + 2 C_2 (Q_2 - 2 P_1)P_1 Q_2 P_{\oo} Q_{\oo}^2 \notag  \\
& \qquad  + 2 (C_{2,4} - C_2 D_2 + B_2 H) P_1 Q_2^2 P_{\oo} Q_{\oo} + A_1 Q_2^3 P_{\oo}^2 +  A_2(Q_2 - P_1) P_1^2 Q_{\oo}^2 \Big{)}\notag  \\
P_{\oo 1} & = -\frac{P_{\oo} (A_1 Q_2 + 2 C_1 Q_{\oo} P_1)}{Q_{\oo} P_1} \label{second-derivatives} \\
Q_{\oo 2} & = -\frac{Q_{\oo} (A_2 P_1 + 2 C_2 P_{\oo} Q_2)}{P_{\oo} Q_2} \notag \\
P_{\oo 3} & = -\frac{P_{\oo}(B_1 Q_2 + (D_1 + E_1) P_1 Q_{\oo}+ (2 Q_2 - P_1) P_{\oo} Q_{\oo})}{P_1 Q_{\oo}} \notag \\
Q_{\oo 4} & = -\frac{Q_{\oo}(B_2 P_1 + (D_2 + E_2) Q_2 P_{\oo}+ (2 P_1 - Q_2) P_{\oo} Q_{\oo})}{Q_2 P_{\oo}}. \notag
\end{align}
This leaves only $P_{33}$ and $Q_{44}$ as undetermined second derivatives of $P$ and $Q$.

We are now ready to set up our exterior differential system.  Let $\widehat{\B} = \B \times \R^{10}$, with coordinates $P$, $Q$, $P_{\oo}$, $P_1$, $P_3$, $Q_{\oo}$, $Q_2$, $Q_4$, $P_{33}$, $Q_{44}$ on the $\R^{10}$ factor.  Let $\W$ be the rank 8 Pfaffian EDS on $\widehat{\B}$ generated by the 1-forms
\begin{align*}
\Theta_1 & = dP - P_{\overline{0}} (\thetabar - Q_2 \theta) - P_1 \theta_1 - P_3 \eta^1, \\
\Theta_2 & = dQ - Q_{\overline{0}} (\thetabar - P_1 \theta) - Q_2 \theta_2 - Q_4 \eta^2,
\end{align*}
and similar forms $\Theta_3, \ldots, \Theta_8$ prescribing conditions \eqref{nmi-dPQ-eqs2} for $dP_{\oo}$, $dP_1$, $dP_3$, $dQ_{\oo}$, $dQ_2$, and $dQ_4$, substituting the values \eqref{second-derivatives} for $P_{\oo \oo}$, $P_{\oo 1}$, $P_{\oo 3}$, $Q_{\oo \oo}$, $Q_{\oo 2}$, $Q_{\oo 4}$.  Integral manifolds of $\W$ satisfying the independence condition
$\theta \& \thetabar \& \theta_1 \& \theta_2 \& \eta^1 \& \eta^2 \& \pi_3 \& \pi_4 \neq 0$
are in one-to-one correspondence with the desired functions $P,Q$ defining a \BT/.

The structure equations for this EDS have the form:
\begin{equation}
\begin{bmatrix} d\Theta_1 \\[0.1in] d\Theta_2 \\[0.1in] d\Theta_3 \\[0.1in] d\Theta_4 \\[0.1in] d\Theta_5 \\[0.1in] d\Theta_6 \\[0.1in] d\Theta_7 \\[0.1in] d\Theta_8 \end{bmatrix} \equiv
- \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & 0 & \Pi_1 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\[0.1in]
0 & 0 & 0 & 0 & 0 & \Pi_2 & 0 & 0 \\[0.1in]
\end{bmatrix} \&
\begin{bmatrix} \theta \\[0.1in] \thetabar \\[0.1in] \theta_1 \\[0.1in] \theta_2 \\[0.1in] \eta^1 \\[0.1in] \eta^2 \\[0.1in] \pi_3 \\[0.1in] \pi_4 \end{bmatrix} +
\begin{bmatrix} \Psi_1 \\[0.1in] \Psi_2 \\[0.1in] \Psi_3 \\[0.1in] \Psi_4 \\[0.1in] \Psi_5 \\[0.1in] \Psi_6 \\[0.1in] \Psi_7 \\[0.1in] \Psi_8
\end{bmatrix}
\mod{\Theta_1, \ldots, \Theta_8},  \label{EDS-structure-eqs}
\end{equation}
where
\[ \left.
\begin{aligned}
\Pi_1 & \equiv dP_{33} \\
\Pi_2 & \equiv dQ_{44}
\end{aligned} \right\} \mod{\theta, \thetabar, \theta_1, \theta_2, \eta^1, \eta^2, \pi_3, \pi_4},
\]
and $\Psi_1, \ldots, \Psi_8$ are 2-forms which are quadratic in the forms $\theta, \thetabar, \theta_1, \theta_2, \eta^1, \eta^2, \pi_3, \pi_4$, with coefficients which are polynomial functions of the two quantities
\begin{align}
& \big{(} 2H P_1(Q_2 - P_1) Q_{\oo}^2 + Q_2(2 C_1 Q_2+ J_1(P_1 - Q_2)) Q_{\oo} + A_1 Q_2^2 \big{)} P_{\oo} \notag \\
& \qquad \qquad \qquad  - 2 C_2 P_1^2 Q_{\oo}^2 + 2 (C_{2,4} - C_2 D_2 + B_2 H) Q_{\oo} P_1 Q_2 , \label{zero4}
\end{align}
\begin{align}
& \big{(} 2H Q_2(P_1 - Q_2) P_{\oo}^2 + P_1(2 C_2 P_1+ J_2(Q_2 - P_1)) P_{\oo} + A_2 P_1^2 \big{)} Q_{\oo} \notag \\
& \qquad \qquad \qquad  - 2 C_1 Q_2^2 P_{\oo}^2 + 2 (C_{1,3} - C_1 D_1 + B_1 H) P_{\oo} P_1 Q_2 . \label{zero5}
\end{align}
In order to find integral manifolds, we must restrict $\W$ to the locus $\Z \subset \widehat{\B} $ defined by the simultaneous vanishing of \eqref{zero4} and \eqref{zero5}.  (These relations could also be found more directly, by computing $d(dP_1) = d(dQ_2) = 0$.)
While relations of this sort could easily lead to hopeless incompatibility for the PDE system given by \eqref{nmi-dPQ-eqs1} and \eqref{nmi-dPQ-eqs2}, it turns out that differentiating these quantities yields no new relations.


A case-by-case analysis, based on the vanishing or non-vanishing of various torsion coefficients of $(\scM, \I)$, shows that the functions $P_{\oo}, Q_{\oo}, P_1, Q_2$ are all nonzero on an open subset $\Z^0 \subset \Z$ which is surjective for the projection $\Z \to \B$.  Since normality requires that these functions be generically nonzero, we further restrict $\W$ to this open subset.

Let $\overline{\W}$ denote the pullback of $\W$ to $\Z^0$;
$\overline{\W}$ is a rank 6 Pfaffian EDS on $\Z^0$.  Because differentiating the equations defining $\Z$ yields no new relations, $\overline{\W}$ is torsion-free; moreover, it is straightforward to check that $\overline{\W}$ is involutive with last nonzero Cartan character $s_1 = 2$. (See \cite{CFB} for a discussion of Cartan's test and involutivity.)
Therefore, it follows from the Cartan-K\"ahler Theorem that local integral manifolds exist and are parametrized by 2 functions of one variable.


We summarize this result as:

\begin{prop}
Let $(\scM, \I)$ be a hyperbolic Monge-Amp\`ere system which is not Monge-integrable,
and which is Darboux-integrable after one prolongation.  Then near
any point $p \in \scM$ there is an open set $U\subset\scM$ around $p$ such that
the restriction of $\I$ to $U$ is linked to the wave equation by a normal \BT/; moreover, the set of all such \BT/s is parametrized by 2 functions of one variable.
Up to contact transformations, all such B\"acklund transformations
preserve the space of independent variables $x,y$.
\end{prop}


\subsection{The Monge-integrable case} \label{MIcase}\label{DtoBmonge}

In this subsection $(\scM, \I)$ is assumed to be a hyperbolic \MA/ system which is Monge-integrable and Darboux-integrable after one prolongation.  As explained below, we will construct a canonical coframing associated to the partial prolongation of $\I$, and then proceed as in \S \ref{NMIcase}.

A similar argument to that of Lemma \ref{NMIm5coframe} can be used to prove:
\begin{lemma}\label{MIm5coframe} Near any point of $\scM$, there exists a coframing $(\theta,\pi_1,\pi_2,\eta^1,\eta^2)$
such that $\theta$ spans the 1-forms of $\I$, and the characteristic systems
$\calC_1, \calC_2$ of $\I$ have
derived flags
$$
\calC_1 = \{ \theta,\pi_1,\eta^1\} \supset \{\pi_1, \eta^1\} \supset \{\eta^1 \} =\calC_1^{(\infty)},
\qquad
\calC_2 = \{ \theta,\pi_2,\eta^2\} \supset \{\pi_2, \eta^2\} =\calC_2^{(\infty)}.
$$
\end{lemma}
Indeed, the same coframing as that given in the proof of Lemma \ref{NMIm5coframe} satisfies the conditions of Lemma \ref{MIm5coframe}.  Note that this lemma only assumes the Monge-integrability of $(\scM, \I)$.

In terms of the local coframing on $\scM$ given by the lemma, the
partial prolongation $(\scM', \I')$ is defined as follows:
let $\scM' = \scM \times \R$, with coordinate $r$ on the $\R$ factor,
and let $\I'$ be the Pfaffian system on $\scM'$
generated by $\theta$, the 1-form $\theta_1 = \pi_1 - r \eta^1$, and the 2-form $\pi_2 \& \eta^2$.


\begin{lemma}\label{MIn6coframe} Near any point of $\scM'$ there exists a coframing
$(\theta,\theta_1,\pi_2,\eta^1,\eta^2,\pi_3)$ such that
$\I'$ is generated by $\theta,\theta_1$, and $\pi_2 \& \eta^2$, satisfying
\begin{equation}\label{mi-Darboux-structure-0}
\begin{aligned}
d\theta & = -\theta_1 \& \eta^1 - \theta_2 \& \eta^2 \mod{\theta} %\notag
\\
d\theta_1 & = -\pi_3 \& \eta^1 \mod{\theta, \theta_1}, %\label{mi-Darboux-structure-0}
\end{aligned}
\end{equation}
with the derived flags of the characteristic systems of $\I'$ given by
\begin{gather*}
\Cprime1  = \{ \theta, \theta_1, \eta^1, \pi_3 \}  \supset
 \{ \theta_1, \eta^1, \pi_3 \} \supset  \{ \eta^1, \pi_3 \}  = \Cprime1^{(\infty)},
\\
\Cprime2  = \{ \theta, \theta_1, \pi_2, \eta^2\} \supset
\{\theta, \pi_2, \eta^2\} \supset  \{\pi_2, \eta^2\} = \Cprime2^{(\infty)}.
\end{gather*}
\end{lemma}

\begin{proof}
As in the proof of Lemma \ref{n7coframe}, the usual or ``full" prolongation
of $\I$ on $\scM \times \R^2$ is generated by $\theta$, $\theta_1$ and
$\theta_2 = \pi_2 - t\, \eta^2$.  We may construct the coframing $(\theta, \theta_1, \theta_2, \eta^1, \eta^2, \pi_3, \pi_4)$ on $\scM \times \R^2$ precisely as in Lemma \ref{n7coframe}, and this coframing satisfies the  structure equations \eqref{nmi-Darboux-structure-0}.


The hypothesis of Darboux-integrability implies that the characteristic system
$$\K_1 = \{ \theta, \theta_1,\theta_2, \eta_1, \pi_3\}$$ of the prolongation contains a rank 2 Frobenius system.  As in Lemma \ref{n7coframe}, it follows from the structure equations and the
construction of $\pi_3$ that
$$\K_1^{(1)} = \{ \theta, \theta_1, \eta_1, \pi_3\}.$$
However, this system is well-defined on $\scM' = \scM \times \R$, and it coincides with
the characteristic system $\Cprime1$ given in the statement of the present Lemma.
It follows that we may adjust $\pi_3$ so that it lies in the rank 2 Frobenius system $\Cprime1^{(\infty)}$.

Note that the second characteristic system $\Cprime2$ is simply the sum of $\I'$ and
the pullback of the characteristic system $\calC_2$ of $\I$, and the structure of its
derived flag follows from that of $\calC_2$.
\end{proof}





The conditions in Lemma \ref{MIn6coframe} are preserved by changes of coframing of the form
\begin{equation} \label{MI-Darboux-freedom}
\begin{bmatrix}
\tilde{\theta}\\[0.1in] \tilde{\theta}_1 \\[0.1in] \tilde{\pi}_2 \\[0.1in] \tilde{\eta}^1 \\[0.1in] \tilde{\eta}^2 \\[0.1in] \tilde{\pi}_3
\end{bmatrix} =
\begin{bmatrix}
c & 0 & 0 & 0 & 0 & 0  \\[0.1in]
0 & a_1 c & 0 & 0 & 0 & 0   \\[0.1in]
0 & 0 & a_2 c & 0 & b_1 & 0   \\[0.1in]
0 & 0 & 0 & a_1^{-1} & 0 & 0   \\[0.1in]
0 & 0 & b_2 & 0 & a_2^{-1} & 0    \\[0.1in]
0 & 0 & 0 & b_3 & 0 & a_1^2 c
\end{bmatrix}^{-1}
\begin{bmatrix}
\theta\\[0.1in] \theta_1 \\[0.1in] \pi_2 \\[0.1in] \eta^1 \\[0.1in] \eta^2 \\[0.1in] \pi_3
\end{bmatrix},
\end{equation}
with $a_1, a_2, c \neq 0$.  Let $G \subset GL(6, \R)$ be the group of such transformations, and let $\calP$ be the $G$-structure on $\scM'$ of which the coframing of Lemma \ref{MIn6coframe} is a section.

After absorbing as much torsion as possible and differentiating to uncover relations among the torsion, $\calP$ has structure equations
\begin{multline}%\hskip -.2in
\begin{bmatrix} d\theta\\[0.1in] d\theta_1\\[0.1in] d\pi_2 \\[0.1in] d\eta^1\\[0.1in] d\eta^2 \\[0.1in] d\pi_3
\end{bmatrix} = -
%\begin{bmatrix}
{\setlength\arraycolsep{0.5pt}
\left[  \begin{array}{c c c c c c}
\gamma & 0 & 0 & 0 & 0 & 0   \\[0.1in]
0 & \ \gamma + \alp_1 & 0 & 0 & 0 & 0  \\[0.1in]
0 & 0 & \gamma + \alp_2 & 0 & \beta_1 & 0  \\[0.1in]
0 & 0 & 0 & -\alp_1 & 0 & 0  \\[0.1in]
0 & 0 & \beta_2 & 0 & -\alp_2 & 0  \\[0.1in]
0 & 0 & 0 & \bet_3 & 0 & \gamma + 2\alp_1
\end{array}\right]
}
% \end{bmatrix}
 \&
\begin{bmatrix} \theta\\[0.1in] \theta_1\\[0.1in] \pi_2 \\[0.1in] \eta^1\\[0.1in] \eta^2 \\[0.1in] \pi_3
\end{bmatrix}
%\\[0.2in]
\label{MI-Darboux-structure}
- \begin{bmatrix}
\theta_1 \& \eta^1 \!  +  \! \pi_2 \& \eta^2 \\[0.1in]
\pi_3 \& \eta^1  \! + \! (A \pi_2 \! + \! B \eta^2) \& \theta \\[0.1in]
0 \\[0.1in]
0 \\[0.1in]
0 \\[0.1in]
C \pi_3 \& \theta_1
\end{bmatrix}.
\end{multline}
Because of the dimensions of the derived flags of the characteristic systems (given in Lemma \ref{MIn6coframe}), $A, B$ are not both zero.  Furthermore, we can choose a local section $\sigma: \scM' \to \calP$ satisfying the conditions that $\eta^1 = dx$, $\eta^2 = dy$, $\pi_2$ is exact, and $\pi_3$ is integrable.
%\begin{align}
%& \eta^1= dx, \qquad \eta^2 = dy, \notag \\
%& \pi_2 \text{\ is exact,} \label{MIcoframe-conds} \\
%& \pi_3 \text{\ is integrable.} \notag
%\end{align}
The resulting coframing is uniquely determined.

When we pull back the structure equations via $\sigma$, the pseudoconnection forms $\alp_1$, $\alp_2$, $\bet_1$, $\bet_2$, $\bet_3$, $\gamma$ become semi-basic.  By making use of the remaining ambiguity in these forms and the conditions imposed thus far on the coframing, we can assume that
%\begin{align*}
$$
\alp_1  = E\eta^1, \qquad
\alp_2  = 0,\qquad
\beta_1  = G \pi_2,\qquad
\beta_2  = 0, \qquad
\beta_3  = (-H + 2E) \pi_3, \qquad
\gamma  = F \pi_2 + G \eta^2
$$
%\end{align*}
for some functions $E,F,G,H$.
Then the structure equations for this coframing become:
\begin{align}
d\theta & = \theta \& (F \pi_2 + G \eta^2)  - \theta_1 \& \eta^1 - \pi_2 \& \eta^2   \notag \\
d\theta_1 & = \theta_1 \& (F \pi_2 + E \eta^1 + G \eta^2)   - \pi_3 \& \eta^1 + \theta \& (A \pi_2 + B \eta^2)   \notag \\
d\pi_2 & = 0 \label{MI-Darboux-section-structure} \\
d\eta^1 & = 0 \notag   \\
d\eta^2 & =  0  \notag \\
d\pi_3 & = \pi_3 \& (-C \theta_1 + F \pi_2 + H \eta^1 + G \eta^2) .    \notag
\end{align}

Once again, we will need to compute relations among the derivatives of the torsion functions in order to show that $(\scM, \I)$ has a \BT/ to the wave equation.  We begin by differentiating the structure equations \eqref{MI-Darboux-section-structure}.
We denote derivatives as, e.g.,
\[ dA = A_{0} \theta + A_{1} \theta_1 + A_{2} \pi_2 + A_{3} \eta^1 + A_{4} \eta^2 + A_{5} \pi_3. \]

Computing $d(d\theta) = d(d\theta_1) = d(d\pi_3) = 0$ yields the following equations for the derivatives of the torsion functions:
\begin{align}
dA & = A_{0} \theta + AC \theta_1 + A_2 \pi_2 - AE \eta^1 + A_4 \eta^2  \notag \\
dB & = B_{0} \theta + BC \theta_1 + A_4 \pi_2 - BE \eta^1 + B_4 \eta^2  \notag \\
dC & = C_1 \theta_1 + CF \pi_2 + C_3 \eta^1 + CG \eta^2 + C_5 \pi_3
\notag \\
dE & = E_1 \theta_1 + 2A \pi_2 + E_3 \eta^1 + 2B \eta^2 - C \pi_3
\label{MI-dtorsion} \\
dF & = AC \theta + F_2 \pi_2 + A \eta^1 + F_4 \eta^2
\notag \\
dG & = BC \theta + F_4 \pi_2 + B \eta^1 + G_4 \eta^2
\notag \\
dH & = (CE - C_3) \theta_1 + A \pi_2 + H_3 \eta^1 + B \eta^2 + H_5 \pi_3 .
\notag
\end{align}

We may obtain further relations among the derivatives of the torsion functions by differentiating equations \eqref{MI-dtorsion}.  Computing $d(dF) \equiv d(dG) \equiv 0$ modulo $\pi_2, \eta^2$ and recalling that $A,B$ cannot vanish simultaneously yields
%\begin{align*}
$$
A_0 = A(C_3 - CE), \qquad
B_0 = B(C_3 - CE), \qquad
C_1 = -C^2, \qquad
C_5 = 0.
$$
%\end{align*}
Then computing $d(dA) \equiv d(dB) \equiv  0$ modulo $\theta, \pi_2, \eta^2$ yields
\[ E_1 = 2(CE - C_3), \]
and $d(dC)=0$ implies that
\[ dC_3 = (C^2E - 2C C_3) \theta_1 + (AC + F C_3) \pi_2 + C_{33} \eta^1 + (BC + G C_3) \eta^2 - C^2 \pi_3. \]
Finally, computing
$d(dA) \equiv d(dB) \equiv  0$ modulo $\pi_2, \eta^2$ yields
\[ C_{33} = C E_3 + E C_3. \]
We now have all the relations that will be needed for the involutivity calculation below.


Now suppose that $\Mbar=\R^5$ carries a \MA/ system $\Ibar$ representing
the wave equation $Z_{XY}=0$, generated algebraically by the contact form
\begin{equation}\label{MI-defthetabar}
\thetabar = dZ - P\, dX - Q\, dY
\end{equation}
and the 2-forms $dP \& dX$ and $dQ \& dY$.  As in \S \ref{NMIcase},
if there were a B\"acklund transformation $\B \subset \scM \times \Mbar$,
then $Z$ would be a local coordinate on the fibers of $\B \to \scM$
and the functions $X, Y, P, Q$ on $\B$ would satisfy the B\"acklund condition
\begin{equation}\label{MI-bcond}
\{dP \& dX, dQ \& dY\} \equiv \{\pi_1 \& \eta^1, \pi_2 \& \eta^2\}
\mod \theta,\thetabar .
\end{equation}

As in \S \ref{NMIcase}, let $\B' = \scM' \times \R$, again with $Z$ as the coordinate on the second factor;
we extend the projection $\scM' \to \scM$ to a projection $\B' \to \B$ by the identity
on the second factor.  We will regard $X, Y, P, Q$ as functions on $\B'$,
but require that
\[ dX, dY, dP, dQ \in \{\theta, \thetabar, \theta_1, \pi_2, \eta^1, \eta^2\} \]
so that they are in fact well-defined on $\B$.
In order to satisfy the \BB/ condition \eqref{MI-bcond}, we will furthermore require that
\begin{equation}
 \{dX, dP\} \subset \{\theta, \thetabar, \theta_1, \eta^1\}, \qquad \{dY, dQ\} \subset \{\theta, \thetabar, \pi_2, \eta^2\}.  \label{MI-inclusion1}
\end{equation}


The same argument as that given in \S \ref{NMIcase} shows that by a contact transformation on $\Mbar$, we may assume that $X=x$.  However, the same is not true for $Y$: the system $\{\theta, \thetabar, \pi_2, \eta^2\}$ on $\B$ contains a rank 3 integrable subsystem, so we cannot necessarily arrange to have $\eta^2 \in \{dY, dQ\}$.  There are three different, geometrically natural conditions that we could impose on the intersection of the rank 2 Pfaffian systems $\{dY, dQ\}$ and $\{\pi_2, \eta^2\}$, each of them potentially leading to a different type of \BT/:
\begin{enumerate}
\item $\{dY, dQ\} \cap \{\pi_2, \eta^2\}$ has rank 1 and is spanned by a non-integrable 1-form.
\item $\{dY, dQ\} \cap \{\pi_2, \eta^2\}$ has rank 1 and is spanned by an integrable 1-form.
\item $\{dY, dQ\} \cap \{\pi_2, \eta^2\}$ has rank 2.
\end{enumerate}
In cases (2) and (3) we can arrange that $Y=y$ via contact transformations on $\scM$ and $\Mbar$, but in case (1) this is not possible.

\subsubsection{Case (1)}
In this case we have
\[ \thetabar = dZ - P\, dx - Q\, dY, \]
and condition \eqref{MI-inclusion1} becomes
\[ dP \in \{\theta, \thetabar, \theta_1, \eta^1\}, \qquad dY, dQ \in \{\theta, \thetabar, \pi_2, \eta^2\}.  \]
Suppose that
\begin{align}
dP &= P_0 \theta + P_{\overline{0}} \thetabar + P_1 \theta_1 + P_3 \eta^1 \notag \\
dQ &= Q_0 \theta + Q_{\overline{0}} \thetabar + Q_2 \pi_2 + Q_4 \eta^2 \label{mi-dP-eqs-case1} \\
dY &= Y_0 \theta + Y_{\overline{0}} \thetabar + Y_2 \pi_2 + Y_4 \eta^2. \notag
\end{align}
Normality of the \BT/ requires that $P_1 \neq 0, \ Q_2 Y_4 - Q_4 Y_2 \neq 0$, and $P_1 \neq Q_2 Y_4 - Q_4 Y_2.$

The argument proceeds in much the same fashion as that of \S \ref{NMIcase}: differentiating equations \eqref{mi-dP-eqs-case1} leads to relations among the derivatives of $P,Q,Y$.  Eventually we are led to a Pfaffian exterior differential system $\W$ whose integral manifolds satisfying the independence condition $\theta \& \thetabar \& \theta_1 \& \pi_2 \& \eta^1 \& \eta^2 \neq 0$ are in one-to-one correspondence with the desired functions $P,Q,Y$ defining a \BT/.  This EDS is involutive with last nonzero Cartan character $s_3 = 1$.
Therefore, local integral manifolds exist and are parametrized by 1 function of three variables.

If we impose the additional condition that the \BT/ be holonomic, we find that the resulting EDS is involutive with last nonzero Cartan character $s_2 = 2$.  Therefore, among the \BT/s of this type, there is a small, proper subset, parametrized by 2 functions of two variables, consisting of holonomic transformations.

\subsubsection{Case (2)}
In this case, we can use contact transformations on $\scM$ and $\Mbar$ to arrange that $\{dY, dQ\} \cap \{\pi_2, \eta^2\}$ is spanned by $\eta^2 = dy = dY$.  Then we have
\[ \thetabar = dZ - P\, dx - Q\, dy, \]
and condition \eqref{MI-inclusion1} becomes
\[ dP \in \{\theta, \thetabar, \theta_1, \eta^1\}, \qquad dQ \in \{\theta, \thetabar, \pi_2, \eta^2\}.  \]
Suppose that
\begin{align}
dP &= P_0 \theta + P_{\overline{0}} \thetabar + P_1 \theta_1 + P_3 \eta^1, \label{mi-dP-eqs-case2} \\
dQ &= Q_0 \theta + Q_{\overline{0}} \thetabar + Q_2 \pi_2 + Q_4 \eta^2. \notag
\end{align}
Normality of the \BT/ requires that $P_1, Q_2 \neq 0$ and $P_1 \neq Q_2.$


Differentiating equations \eqref{mi-dP-eqs-case2} leads to relations among the derivatives of $P,Q$, and to a Pfaffian exterior differential system $\W$ whose integral manifolds satisfying the independence condition $\theta \& \thetabar \& \theta_1 \& \pi_2 \& \eta^1 \& \eta^2 \neq 0$ are in one-to-one correspondence with the desired functions $P,Q$ defining a \BT/.  This EDS is involutive with last nonzero Cartan character $s_2 = 1$.
Therefore, local integral manifolds exist and are parametrized by 1 function of two variables.

If we impose the additional condition that the \BT/ be holonomic, we find that the resulting EDS is involutive with last nonzero Cartan character $s_1 = 3$.  Therefore, among the \BT/s of this type, there is a small, proper subset, parametrized by 3 functions of one variable, consisting of holonomic transformations.


\subsubsection{Case (3)}
In this case, we can use contact transformations on $\scM$ and $\Mbar$ to arrange that $\eta^2 = dy = dY, \ \pi_2 = dQ$.  Then we have
\[ \thetabar = dZ - P\, dx - Q\, dy, \]
and condition \eqref{MI-inclusion1} becomes
\[ dP \in \{\theta, \thetabar, \theta_1, \eta^1\}.  \]
Suppose that
\begin{equation}
dP = P_0 \theta + P_{\overline{0}} \thetabar + P_1 \theta_1 + P_3 \eta^1. \label{mi-dP-eqs-case3}
\end{equation}
Normality of the \BT/ requires that $P_1 \neq 0$ and $P_1 \neq 1$.

Differentiating equation \eqref{mi-dP-eqs-case3} leads to relations among the derivatives of $P$, and to a Pfaffian exterior differential system $\W$ whose integral manifolds satisfying the independence condition $\theta \& \thetabar \& \theta_1 \& \pi_2 \& \eta^1 \& \eta^2 \neq 0$ are in one-to-one correspondence with the desired functions $P$ defining a \BT/.

The involutivity calculation in this case depends on the torsion functions in the structure equations \eqref{MI-Darboux-section-structure}.  If
\begin{equation}
 AG - BF = AC + F(CE - C_3) = AF^2 + F A_2 - A F_2 = BF^2 + F A_4 - A F_4 = 0, \label{mi-case3-cond}
\end{equation}
then $\W$ is involutive with last nonzero Cartan character $s_2 = 1$, and so local integral manifolds exist and are parametrized by 1 function of two variables.   Otherwise, there are no solutions with $P_1 \neq 0$, and hence no normal \BT/s of this type.

Observe that in this case, the $G$-structure on the B\"acklund transformation $\B$ (cf. \S \ref{wsetup}) will satisfy the condition that (omitting obvious pullback notations)
\[ \{\om^3, \om^4\} = \{\pi_2, \eta^2\} = \{dY, dQ\}. \]
Therefore, all transformations of this type satisfy the hypotheses of Proposition \ref{one-C-vector-zero-prop} and so are holonomic.

We summarize these results as:

\begin{prop}
Let $(\scM, \I)$ be a hyperbolic Monge-Amp\`ere system which is Monge-integrable, and Darboux-integrable after one prolongation.  Then there exist B\"acklund transformations  of types (1) and (2) above between $(\scM, \I)$ and the standard wave equation $(\Mbar, \Ibar)$, and of type (3) if the torsion functions of $(\scM, \I)$ satisfy \eqref{mi-case3-cond}.
The generic B\"acklund transformation is of type (1) and does not preserve the space of independent variables.  There are both holonomic and non-holonomic \BT/s of types (1) and (2), and all B\"acklund transformations of type (3) are holonomic.
\end{prop}








