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