  In this paper, we study two different classes of normal geodesic flows
corresponding to the left-invariant sub-Riemannian metric on the
$(2n+1)$-dimensional Heisenberg group. The first class corresponds to the
left-invariant distribution, while the second corresponds to the
right-invariant one. We prove that corresponding Hamiltonian L-L and L-R
systems are completely integrable.
