Algebraic integrability of confluent Neumann system

In this paper we study the Neumann system, which describes the harmonic oscillator (of arbitrary dimension) constrained to the sphere. In particular we will consider the confluent case where two eigenvalues of the potential coincide, which implies that the system has $S^1$ symmetry. We will prove complete algebraic integrability of the confluent Neumann system and show that its flow can be linearized on the generalized Jacobian torus of some singular algebraic curve. The symplectic reduction of S 1 action will be described and we will show that the general Rosochatius system is a symplectic quotient of the confluent Neumann system, where all the eigenvalues of the potential are double