Complete integrability of subriemannian geodesic flows on $\mathbb{S}^7

Four subriemannian (SR) structures over the Euclidean sphere $\mathbb{S}^7$ are considered in accordance to the previous literature. The defining bracket generating distribution is chosen as the horizontal space in the Hopf fibration, the quaternionic Hopf fibration or spanned by a suitable number of canonical vector fields. In all cases the induced SR geodesic flow on $T^*\mathbb{S}^7$ is studied. Adapting a method by A. Thimm, a maximal set of functionally independent and Poisson commuting first integrals are constructed, including the corresponding SR Hamiltonian. As a result, the complete integrability in the sense of Liouville is proved for the SR geodesic flow. It is observed that these first integrals arise as the symbols of commuting second order differential operators one of them being a (not necessarily intrinsic) sublaplacian. On the way one explicitly derives the Lie algebras of all SR isometry groups intersected with $O(8)$.