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

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.