INTEGRABLE SYSTEMS ON THE SPHERE WITH ELAST IC INTERACTION POTENTIALS

1. The classical example of a system soluble by the method of separation of variables using elliptic spherical coordinates (elliptische Kugel coordinaten) is the Neumann problem on the motion of 'a point along the n-dimensional sphere S ' ~ = {(x,x) = I} , x = (xo,xl...,xn) E R n+1 (1) in a force field with the potential energy 1 V = ~(Ax,x) , (2) where A: R n+l--+ R n+l is a symmetric linear operator. The proof of this fact can be found, for example, in [1]. The Neumann system is sometimes called an anisotropic harmonic oscillator on the sphere [2]. This name appears to be related to the fact that if restriction (1) is absent, then the system =-OV/Oz with the positively defined potential (2) can be reduced to a system of disconnected harmonic oscillators. In fact, as we shall see from what follows, an analog of the Hooke potential on the sphere is a function of the for