Subriemannian geometry on rank 2 Carnot groups

We study nilpotent left-invariant sub-Riemannian structures with the growth vectors (2,3,4), (2,3,5), and (2,3,5,8).
For the growth vector (2,3,4), i.e., for the left-invariant SR structure on the Engel group, we prove the cut time is equal to the first Maxwell time corresponding to discrete symmetries (reflections) of the exponential mapping. For the growth vector (2,3,5), i.e., for the left-invariant SR structure on the Cartan group, the same fact is a conjecture supported by mathematical and numerical evidence.
For the growth vector (2,3,5,8), we study integrability of the normal Hamiltonian vector field $\vec{H}$. We compute 10 independent integrals of $\vec{H}$, of which only 7 are in involution. After reduction by 4 Casimir functions, the vertical subsystem of $\vec{H}$ (on the dual to the Lie algebra of the 8-dimensional nilpotent Lie algebra) shows numerically a chaotic dynamics, which leads to a conjecture on non-integrability of $\vec{H}$ in the Liouville sense.