  The relation between solutions to Helmholtz's equation on the sphere
$S^{n-1}$ and the $[{\gr sl}(2)]^n$ Gaudin spin chain is clarified. The joint
eigenfuctions of the Laplacian and a complete set of commuting second order
operators suggested by the $R$--matrix approach to integrable systems, based on
the loop algebra $\wt{sl}(2)_R$, are found in terms of homogeneous polynomials
in the ambient space. The relation of this method of determining a basis of
harmonic functions on $S^{n-1}$ to the Bethe ansatz approach to integrable
systems is explained.
