On a class of integrable systems of Monge-Ampère type

Abstract: We investigate a class of multi-dimensional two-component systems of Monge-Amp`ere type that can be viewed as generalisations of heavenly-type equations appearing in self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of skew-symmetric matrix pencils, a classification of normal forms of such systems is obtained. All two-component systems of Monge-Amp`ere type turn out to be integrable, and can be represented as the commutativity conditions of parameter-dependent vector fields. Geometrically, systems of Monge-Amp`ere type are associated with linear sections of the Grassmannians. This leads to an invariant differential-geometric characterisation of the Monge-Amp`ere property.