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We study pairs of structures, such as the Poisson-Nijenhuis structures, onthe tangent bundle of a manifold or, more generally, on a Lie algebroid or aCourant algebroid. These composite structures are defined by two of thefollowing, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, withsuitable compatibility assumptions. We establish the relationships between suchcomposite structures. We then show that the non-degenerate Monge-Amp\`erestructures on 2-dimensional manifolds satisfying an integrability conditionprovide numerous examples of such structures, while in the case of3-dimensional manifolds, such Monge-Amp\`ere operators give rise to generalizedcomplex structures or generalized product structures on the cotangent bundle ofthe manifold.

Compatible structures on Lie algebroids and Monge-Amp\`ere operators