On the integrability of a class of Monge-Ampère equations

We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge-Ampère equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge-Ampère equations. Local as well nonlocal conserved densities are obtained