In this work we study the Goursat distribution and a dynamical system associated with it (see [1], [2]). This sub-Riemannian model has many physical applications. The corresponding geodesic flow turned out to be completely integrable in the Liouville sense. All the first integrals are found explicitly and the corresponding level surfaces are constructed. There is an important and interesting problem to find closed periodic solutions. Some examples of such solutions are constructed.
[1] R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, P.A. Griffiths. Exterior differential systems. Mathematical Sciences Research Institute Publications, 1980.
[2] R. Montgomery. A tour of subriemannian geometries, their geodesics and applications. AMS, 2002.