On a class of integrable systems of Monge-Ampére type

Abstract

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