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We study integrable non-degenerate Monge-Ampere equations of Hirota type in4D and demonstrate that their symmetry algebras have a distinguished gradedstructure, uniquely determining the equations. This is used to deform theseheavenly type equations into new integrable PDE of the second order with largesymmetry pseudogroups. We classify the obtained symmetric deformations anddiscuss self-dual hyper-Hermitian geometry of their solutions, which encodeintegrability via the twistor theory.

Integrable dispersionless PDE in 4D, their symmetry pseudogroups and deformations