We consider the local equivalence problem for the class of linear second order hyperbolic equations in two independent variables under an action of the pseudogroup of contact transformations.É. Cartan's method is used for finding the Maurer -Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants.

We study integrable non-degenerate Monge-Ampère 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.

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We consider the four-dimensional integrable Martínez Alonso-Shabat equation, and list three integrable three-dimensional reductions thereof. We also present a four-dimensional integrable modified Martínez Alonso-Shabat equation together with its Lax pair.We also construct an infinite hierarchy of commuting nonlocal symmetries (and not just the shadows, as it is usually the case in the literature) for the Martínez Alonso-Shabat equation.