Classical r-matrices of rational, trigonometric and elliptic type may be used to define polynomial families of multi-Hamiltonian structures on loop algebras and loop groups corresponding, respectively, to linear (Lie-Poisson) and quadratic (Poisson-Lie/Sklyanin) brackets. The standard r-matrix theory implies commutativity and complete integrability of the flows generated by spectral invariants on finite dimensional Poisson submanifolds consisting of meromorphic Lax matrices over the base curve (of genus 0 or 1) with given pole divisor. The spectral transform yields an identification with the space consisting of pairs of: spectral curves and sheaves supported on them, on which a natural family of algebro-geometric Poisson structures is defined.

A third way to view such Poisson spaces is by identification with symmetric products of a holomorphic Poisson surface with itself, leading to a separation of variables of the flows generated by spectral invariants in the associated ”spectral Darboux coordinates”. It is shown that the generalized Gel’fand-Zakharevich commuting invariants associated to the multi-Hamiltonian structures are the same as the spectral invariants derived from the r-matrix theory, and the “Nijenhuis-Darboux” coordinates given by the eigenvalues and eigenvectors of the Nijenhuis tensor coincide with the spectral Darboux coordinates. Examples illustrating these results include: constrained oscillators in n-dimensions, classical spin systems, Toda and Volterra lattices and reduced systems associated with commuting flows on the stationary manifolds of the NLS, KdV and Boussinesq hierarchies.