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

In this study, we investigate two distinct classes of normal geodesic flows associated with the left-invariant sub-Riemannian metric on the (2n + 1)-dimensional Heisenberg group. The first class arises from the left-invariant distribution, whereas the second is derived from the right-invariant distribution. It is established that the Hamiltonian system corresponding to the left-left (LL) configuration is completely integrable in the non-commutative sense. We demonstrate that the left-right (LR) systems exhibit non-commutative integrability in dimensions exceeding 5, while in dimensions 3 and 5, integrability is achieved in the commutative sense.