A geometric approach to the separability of the Neumann–Rosochatius system

We study the separability of the Neumann–Rosochatius system on the ndimensional sphere using the geometry of bi-Hamiltonian manifolds. Its well-known separation variables are recovered by means of a separability condition relating the Hamiltonian with a suitable (1,1) tensor field on the sphere. This also allows us to iteratively construct the integrals of motion of the system.