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For the rational, elliptic and trigonometric r-matrices, we exhibit the linksbetween three "levels" of Poisson spaces: (a) Some finite-dimensional spaces ofmatrix-valued holomorphic functions on the complex line; (b) Spaces of spectralcurves and sheaves supported on them; (c) Symmetric products of a surface. Wehave, at each level, a linear space of compatible Poisson structures, and themaps relating the levels are Poisson. This leads in a natural way to Nijenhuiscoordinates for these spaces. At level (b), there are Hamiltonian systems onthese spaces which are integrable for each Poisson structure in the family, andwhich are such that the Lagrangian leaves are the intersections of thesymplective leaves over the Poisson structures in the family. Specific examplesinclude many of the well-known integrable systems.Comment: 26 pages, Plain Te

Multi-Hamiltonian structures for r-matrix systems