Title: Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property,
Abstract: We prove that integrability of a dispersionless Hirota type equation implies
the symplectic Monge-Ampere property in any dimension $\geq 4$. In 4D this
yields a complete classification of integrable dispersionless PDEs of Hirota
type through a list of heavenly type equations arising in self-dual gravity. As
a by-product of our approach we derive an involutive system of relations
characterising symplectic Monge-Ampere equations in any dimension.
Moreover, we demonstrate that in 4D the requirement of integrability is
equivalent to self-duality of the conformal structure defined by the
characteristic variety of the equation on every solution, which is in turn
equivalent to the existence of a dispersionless Lax pair. We also give a
criterion of linerisability of a Hirota type equation via flatness of the
corresponding conformal structure, and study symmetry properties of integrable
equations.