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We prove that integrability of a dispersionless Hirota type equation impliesthe symplectic Monge-Ampere property in any dimension $\geq 4$. In 4D thisyields a complete classification of integrable dispersionless PDEs of Hirotatype through a list of heavenly type equations arising in self-dual gravity. Asa by-product of our approach we derive an involutive system of relationscharacterising symplectic Monge-Ampere equations in any dimension.  Moreover, we demonstrate that in 4D the requirement of integrability isequivalent to self-duality of the conformal structure defined by thecharacteristic variety of the equation on every solution, which is in turnequivalent to the existence of a dispersionless Lax pair. We also give acriterion of linerisability of a Hirota type equation via flatness of thecorresponding conformal structure, and study symmetry properties of integrableequations.

Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property