Gaudin models and bending flows: a geometrical point of view

In this paper, we discuss the bi-Hamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl (r)-valued 'spins'. We only consider the classical case, using recent results concerning the quantum models as guiding principles. In particular, we focus on the so-called homogeneous XXX models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the 'standard' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide with the Hamiltonians of the 'bending flows' in the moduli space of polygons in the Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the sl(2) case