Invariant Monge-Amp\`ere equations on contactified para-K\"ahler manifolds

We develop a method for describing invariant Monge-Amp\`ere equations in the sense of V. Lychagin and T. 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 = AdG Z ⊂ g, 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.