  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 $O(8)$.
