# On a class of integrable systems of Monge-Amp\`ere type

## TL;DR

This paper classifies and analyzes a broad class of multi-dimensional Monge-Ampère type systems, revealing their integrability and geometric structure, and connecting them to classical geometric objects like Grassmannians.

## Contribution

It provides a classification of two-component Monge-Ampère systems using Jordan-Kronecker theory and demonstrates their integrability and geometric interpretation.

## Key findings

- All two-component systems are integrable.
- Systems are characterized as commutativity conditions of vector fields.
- They correspond to linear sections of Grassmannians.

## 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.