The Monge-Ampere equation

was studied extensively in the 1800's most notably by Lie. He

showed that it was possible to find first integrals to this

equation. If one solves

for r and t and substitutes into our PDE and separate wrt s then

we obtain the Monge equations

Considering a linear combination of these two shows that they can

be factored if are two real distinct solutions of

and further, the factors are

In your case where then we have which has factors . Thus, we have

which gives rise to

or

leading to the first integrals

and

Sophus Lie (1877) showed that via a contact transformation that it

possible to transform PDE like yours to the wave equation. In

your case the transformation is

giving With the general solution as , gives the general solution of your PDE via (1).