  In this paper we study evolutive first order Mean Field Games in the
Heisenberg group; each agent can move in the whole space but it has to follow
"horizontal" trajectories which are given in terms of the vector fields
generating the group and the kinetic part of the cost depends only on the
horizontal velocity. The Hamiltonian is not coercive in the gradient term and
the coefficients of the first order term in the continuity equation may have a
quadratic growth at infinity. The main results of this paper are two: the
former is to establish the existence of a weak solution to the Mean Field Game
systems while the latter is to represent this solution following the Lagrangian
formulation of the Mean Field Games. We also provide some generalizations to
Heisenberg-type structures.
