Abstract
We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the Sklyanin bracket, and use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet, the open and periodic Toda lattices, the discrete self-trapping model and the Goryachev–Chaplygin gyrostat.
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