Separable coordinates, integrability and the Niven equations

Finite (polynomial) solutions of Laplace's equation are investigated. The unifying features of this study are the so-called Niven equations which yield the dimension of the space of such solutions. In carrying out this study complete sets of solutions are obtained on the n-dimensional sphere in terms of ellipsoidal coordinates. This corresponds to an integrable system having all the integrals of motion given by quadratic orbits of the universal enveloping algebra of O(n+1). They call this system the n-dimensional Euler top. The spectrum of the integrals of motion has been recently computed for n=3 by Komarov and Kuznetsov (1991) using results originally due to Niven (1891). These calculations are extended to arbitrary dimension.