  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.
