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We study second-order PDEs in 4D for which the conformal structure defined bythe characteristic variety of the equation is half-flat (self-dual oranti-self-dual) on every solution. We prove that this requirement implies theMonge-Ampere property. Since half-flatness of the conformal structure isequivalent to the existence of a nontrivial dispersionless Lax pair, our resultexplains the observation that all known scalar second-order integrabledispersionless PDEs in dimensions four and higher are of Monge-Ampere type.Some partial classification results of Monge-Ampere equations in 4D withhalf-flat conformal structure are also obtained.

Second-order PDEs in 4D with half-flat conformal structure