  A variant of the classical optimal transportation problem is: among all joint
measures with fixed marginals and which are dominated by a given density, find
the optimal one. Existence and uniqueness of solutions to this variant were
established in \cite{KM11}. In the present manuscript, we expose an unexpected
symmetry leading to the first explicit examples in two and more dimensions.
These are inspired in part by simulations in one dimension which display
singularities and topology and in part by two further developments: the
identification of all extreme points in the feasible set, and a new approach to
uniqueness based on constructing feasible perturbations.
