  The $n$ integrals in involution for the motion on the $n$-dimensional
ellipsoid under the influence of a harmonic force are explicitly found. The
classical separation of variables is given by the inverse momentum map. In the
quantum case the Schr\"odinger equation separates into one-dimensional
equations that coincide with those obtained from the classical separation of
variables. We show that there is a more general orthogonal parametrisation of
Jacobi type that depends on two arbitrary real parameters. Also if there is a
certain relation between the spring constants and the ellipsoid semiaxes the
motion under the influence of such a harmonic potential is equivalent to the
free motion on the ellipsoid.
