General solutions of the Monge-Amp\`{e}re equation in $n$-dimensional   space
  It is shown that the general solution of a homogeneous Monge-Amp\`{e}re equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system   \frac {\partial \xi_{j}}{\partial x_0}+\sum_{k=1}^{n-1} \xi_{k} \frac {\partial \xi_{j}}{\partial x_{k}}=0 \label{1}   Using the explicit form of solution of this system it is possible to construct the general solution of the Monge-Amp\`{e}re equation. 
D. B. Fairlie and A. N. Leznov