  In this paper we study the geodesic flow on nilmanifolds equipped with a
left-invariant metric. We write the underlying definitions and find general
formulas for the Poisson involution. As an example we develop the Heisenberg
Lie group equipped with its canonical metric. We prove that a family of first
integrals giving the complete integrability can be read off at the Lie algebra
of the isometry group. We also explain the complete integrability on compact
quotients and for any invariant metric.
