We consider a left-invariant sub-Riemannian problem of Engel type on the central extension of the special linear group. Interest in this problem comes from the fact that it has strictly abnormal trajectories, and the normal geodesic flow is Liouville integrable. An extremal trajectory is called strictly abnormal if it is not present among normal geodesics. It is known that the most complicated singularities of the sub-Riemannian metric arise near abnormal trajectories. The presence of a strictly abnormal trajectory in combination with the integrability of the normal geodesic flow makes the problem under consideration a model example for studying the singularities of the sub-Riemannian metric. We apply to the problem the invariant formulation of Pontryagin maximum principle (PMP), in which the vertical subsystem (for adjoint variables) of the Hamiltonian system of PMP is independent of the state variables. We show that the vertical subsystem is reduced to the equation of a skewed pendulum. The first integrals of the system are found and an explicit solution is obtained in a special case. In the general case, we carry out a qualitative analysis of the phase flow of the Hamiltonian system.
Moscow, ICS RAS