  In this paper we study evolutive first order Mean Field Games in the
Heisenberg group~$\He^1$; each agent can move only along "horizontal"
trajectories which are given in terms of the vector fields generating~$\He^1$
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 system while the latter is
to represent this solution following the Lagrangian formulation of the Mean
Field Games.We shall tackle both the Heisenberg-periodic and the non periodic
case following two different approaches. To get these results, we prove some
properties which have their own interest: uniqueness results for a second order
Fokker-Planck equation and a probabilistic representation of the solution to
the continuity equation.\end{abstract}
