  We provide a representation of the weak solution of the continuity equation
on the Heisenberg group $\mathbb H^1$ with periodic data (the periodicity is
suitably adapted to the group law). This solution is the push forward of a
measure concentrated on the flux associated with the drift of the continuity
equation. Furthermore, we shall use this interpretation for proving that weak
solutions to first order Mean Field Games on $\mathbb H^1$ are also mild
solutions.
