Classification of the geodesic curves of the sub-Riemannian LR system on the Heisenberg group in dimension 5

We study the geodesic flow corresponding to the left-invariant sub-Riemannian metric and the right-invariant distribution on the second Heisenberg group. The corresponding Hamiltonian system is completely integrable and in this paper we study its solutions. We obtain a complete classification of the geodesic curves. Moreover, we compute $r(t)$ as the first step of the quadrature.