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 \(M = G/H = \textrm{Ad}_G Z \subset \mathfrak {g}\), where M is the adjoint orbit of a splittable closed element Z of the Lie algebra \(\mathfrak {g}= {{\,\textrm{Lie}\,}}(G)\). The method is then applied to a ten-dimensional semisimple orbit M of the exceptional Lie group \(\textsf{G}_2\) and a complete list of mutually non-equivalent MAEs on N is obtained.