  We study integrable non-degenerate Monge-Ampere equations of Hirota type in
4D and demonstrate that their symmetry algebras have a distinguished graded
structure, uniquely determining the equations. This is used to deform these
heavenly type equations into new integrable PDE of the second order with large
symmetry pseudogroups. We classify the obtained symmetric deformations and
discuss self-dual hyper-Hermitian geometry of their solutions, which encode
integrability via the twistor theory.
