  We adapt the problem of continuous congested optimal transport to the
Heisenberg group with a sub-riemannian metric: we restrict the set of
admissible paths to the absolutely continuous curves which are also horizontal.
We get the existence of equilibrium configurations, known as Wardrop
Equilibria, through the minimization of a convex functional over a suitable set
of measures. To prove existence of such minima, that turn out to be equilibria,
we prove the existence of summable transport densities. Moreover, such
equilibria induces transport plans that solve a Monge-Kantorovic problem
associated with a cost function, depending on the congestion itself, which we
rigorously define.
