A family of the Poisson brackets compatible with the Sklyanin bracket

We introduce a family of compatible Poisson brackets on the space of 2 x 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