On the step-$2$ nilpotent $(n, n(n + 1)/2)$ sub-Riemannian structures

We consider the Sub-Riemannian (SR) problem on the step-2 free nilpotent Lie groups $\mathbb{G}_n$. This problem is classical in SR geometry and in some sense the simplest open problem nowadays. Although the problem satisfies to a wide group of symmetries, the cut locus is known only in the cases of small dimensions $n = 2,3$. In the general case there exists a conjecture by Rizzi–Serres claiming that the cut locus consists on stable points of the specific symmetry. In this paper, we derive the geodesic equations via PMP and study the symmetries of the corresponding Hamiltonian system. Then, using method of reduction over the symmetries, we propose an idea to prove the conjecture for the general $n\geq2$. We study the cases $n = 2,3,4$ in details and show pictures (for $n=2,3$) of the SR wave front with indicated the cut locus on it. (In Russian).