Poisson structures and integrable systems connected with graphs

Completely integrable systems related with graphs of a specific type are studied by the $r$-matrix method. The phase space of such a system is the space of connections on a graph. The nonlinear equations under consideration are Hamiltonian with respect to the Poisson bracket depending on the geometry of the graph and other structures. It is essential that the Poisson bracket be nonultralocal. An involute family of motion integrals is constructed. Explicit formulas for solutions of evolution equations are obtained in terms of solutions of a factorization problem. In the case of the group of loops, a polynomial anzatz for the Lax operator compatible with the Poisson bracket is constructed. Bibliography: 8 titles.