This talk deals with the geodesic flow of the seven-dimensional sphere with respect to four subriemannian structures, which are given by the Hopf fibration, the quaternionic Hopf fibration, and the two distributions spanned by a certain number of canonical vector fields.

Using the method by Thimm that appeared in the beginning of 1980's, one can prove the complete integrability in the sense of Liouville for the geodesic flows associated to each of the four subriemannian structures, by constructing the explicit first integrals.

The differential operators associated with the constructed first integrals are also studied.

The talk is based on a collaboration with Wolfram Bauer and Abdellah Laaroussi.