the compatibility conditions of the system (\[eq:ms:12\]).

Example. Nonlinear interaction of waves and particles. {#sec:7}
======================================================

Let us consider the inhomogeneous system describing the propagation of shock waves intensity in nonlinear interaction of waves and particles [@Luneburg:1964]

\[eq:e1:1\] u\_x+\_y=2\^[1/2]{}a (u/2)(/2),u\_y-\_x=-2\^[1/2]{}a(u/2)(/2).

It should be noted that the compatibility condition of the mixed derivatives of $\phi$ corresponds to the Liouville equation

\[eq:e1:2\] u\_[xx]{}+u\_[yy]{}=a\^2.

Therefore, each solution of the system (\[eq:e1:1\]) also gives us a solution of the Liouville equation (\[eq:e1:2\]). We use the methods presented in Section \[sec:6\] to obtain the general solution of the system (\[eq:e1:1\]). First, write the system (\[eq:e1:1\]) in the matrix form

\[eq:e1:3\]

(

[c c]{} 1 & 0\
0 & -1

)

(

[c ]{} u\_x\
\_x

)

+

(

[c c]{} 0 & 1\
1 & 0

)

(

[c]{} u\_y\
\_y

)

=

(

[c]{} b\_1\
b\_2

)

,

where $$b_1=2^{1/2}a \exp(u/2)\sin(\phi/2),\qquad b_2=2^{1/2}a\exp{u/2}\cos(\phi/2).$$The matrices $\mathcal{A}^i$ introduced in equation (\[eq:ms:4\]) are given by $$\mathcal{A}^1={\left( \begin{array}{c c}
1 & 0\\
0 & -1
\end{array} \right)},\qquad \mathcal{A}^2={\left( \begin{array}{c c}
0 & 1\\
1 & 0
\end{array} \right)}.$$Condition (\[eq:ms:8\]) is then satisfied by the scalar function $\Omega$, the wave vector $\lambda$ and the rotation matrix $L$, defined by $$\Omega=12^{1/4}(1-\epsilon^i),\quad \lambda=(1,i),\quad L={\left( \begin{array}{c c} l_{11} & l_{12}\\
l_{21} & l_{22}\end{array} \right)},\quad \epsilon=\pm 1,$$together with their complex conjugates, where $$\begin{aligned}
l_{11}&=l_{22}=-108^{1/4}i{\left( \frac{3^{1/2}\epsilon (b_1+ib_2)^2+i(b_1-ib_2)^2}{6(1-\epsilon i) (b_1^2+b_2^2)} \right)}\\
l_{12}&=-l_{21}=108^{1/4}i{\left( \frac{3^{1/2}\epsilon i (b_1+ib_2)^2+(b_1-ib_2)^2}{6(1-\epsilon i) (b_1^2+b_2^2)} \right)}
\end{aligned}$$Since the wave vectors $\lambda$ and $\bar{\lambda}$ are constant, 