Multi-Hamiltonian structure of self-dual gravity

We discover multi-Hamiltonian structure of the complex MongeAmpère equation (CMA) set in a real first-order two-component form. Therefore, by Magri’s theorem this is a completely integrable system in four real dimensions. We start with Lagrangian and Hamiltonian densities in real variables, a symplectic form and the Hamiltonian operator, that determines the Poisson bracket. We have calculated all point symmetries of the two-component CMA system and Hamiltonians of the symmetry flows. We have found two new real recursion operators for symmetries which commute with the operator of a symmetry condition on solutions of the CMA system. These two couples of operators form two Lax pairs for the two-component system. The recursion operators, being applied to the first Hamiltonian operator, generate infinitely many real Hamiltonian structures. We show how to construct an infinite hierarchy of higher commuting flows together with the corresponding infinite chain of their Hamiltonians.