Sub-Riemannian Geodesics on the 3D Heisenberg Nilmanifold

First, we describe the dynamical properties of the geodesic flow for $M$: periodic and dense orbits, the dynamical characteristic of the normal Hamiltonian flow of Pontryagin's maximum principle, and its integrability properties. We show that it is integrable in the Liouville sense on a non-zero level hypersurface $\Sigma$ of the Hamiltonian outside of some smaller eigen-hypersurface in $\Sigma$ and does not possess non-trivial analytic integrals on the entire $\Sigma$. We then obtain precise upper and lower bounds for sub-Riemannian balls and distances in $G$, and based on this, we estimate the cut time for sub-Riemannian geodesics in $M$.