On the integrability of a third-order Monge-Ampère type equation

A third-order equation, similar to a Monge-Ampère equation, is studied, this being achieved by first converting the equation to a 3 × 3-hydrodynamics system. A Lax pair for this hydrodynamic system is constructed and the integrability of this system established. In the degenerate case the equations reduce to a simpler integrable hydrodynamic equation and this case is also studied. The interpretation of this equation as a higher-order Monge-Ampère equation leads to various generalizations which are discussed briefly