Lie solutions of Riemannian transport equations on compact manifolds

Given a couple of smooth positive measures of same total mass on a compact Riemannian manifold, the associated optimal transport equation admits a symplectic Monge-Ampère structure, hence Lie solutions (in a restricted sense, though, still expressing measure-transport). Properties of such solutions are recorded; a structure result is obtained for regular ones (each consisting of a closed 1-form composed with a diffeomorphism) and a quadratic cost-functional proposed for them.