Invariant Monge-Ampère equations on contactified para-Kähler manifolds

We develop a method for describing invariant PDEs of Monge-Ampère type in the sense of Lychagin and Morimoto (MAE) on a homogeneous contact manifold N of a semisimple Lie group G, which is the contactification of the homogeneous symplectic manifold M=G/H=AdGZ, where M is the adjoint orbit of a splittable closed element Z of the Lie algebra g=Lie(G). The method is then applied to a ten-dimensional semisimple orbit M of the exceptional Lie group G2 and a complete list of mutually non-equivalent MAEs on N is obtained.