Finite-zone potentials and integrable systems on a sphere with quadratic potential (Q1063760)

\textit{C. Neuman} [J. Reine Angew. Math. 56, 46-63 (1859)] integrated the system describing the movement of particles on the unit sphere \(S^ n\) under the action of the quadratic potential \(u(x)=\sum^{n}_{i=0}a_ ix^ 2_ i\) for \(n=2\). Uhlenbeck and Devani recently stated explicitly the set of algebraic integrals in the involution which totally ensures integrability in the general case. We state this, following a work of \textit{J. Moser} [preprint, Courant. Inst. (1978)].