  This paper is concerned with the study of the Monge optimal transport problem
in sub-Riemannian manifolds where the cost is given by the square of the
sub-Riemannian distance. Our aim is to extend previous results on existence and
uniqueness of optimal transport maps to cases of sub-Riemannian structures
which admit many singular minimizing geodesics. We treat here the case of
sub-Riemannian structures of rank two in dimension four.
