  We consider the $n$-dimensional Chaplygin sphere under the assumption that
the mass distribution of the sphere is axisymmetric. We prove that for initial
conditions whose angular momentum about the contact point is vertical, the
dynamics is quasi-periodic. For $n=4$ we perform the reduction by the
associated $SO(3)$ symmetry and show that the reduced system is integrable by
the Euler-Jacobi theorem.
