Integrability of the sub-Riemannian geodesic flow of the left-invariant metric on the Heisenberg group

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.