Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups

This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3,6,14)$, the other is defined by two generatrices and growth vector $(2,3,5,8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.