  In this article we use algebro-geometric tools to describe the structure of a
superintegrable system. We study degenerate Neumann system with potential
matrix that has some eigenvalues of multiplicity greater than one. We show that
the degenerate Neumann system is superintegrable if and only if its spectral
curve is reducible and that its flow can be linearized on the generalized
Jacobians of the spectral curve. We also show that the generalized Jacobians of
the hypereliptic component of the spectral curves are models for the minimal
invariant tori of the flow. Moreover the spectral invariants generate local
actions that span the invariant tori, while the moment maps for the rotational
symmetries provide additional first integrals of the system. Using our results
we reproduce already known facts that the degenerate Neumann system is
superintegrable, if its potencial matrix has eigenvalues of multiplicity
greater or equal than $3$.
