Compatible nonlocal Poisson brackets

In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a special type for compatible nonlocal Poisson brackets of hydrodynamic type is solved. The compatible pairs of nonlocal Poisson brackets of hydrodynamic type have a more simple description in special coordinates in which the metrics corresponding to these brackets are diagonal (see [1], [2]), but for an effective construction of the hierarchies we need a different approach developed in this paper. For compatible local Poisson brackets of hydrodynamic type (the Dubrovin–Novikov brackets [3]), the corresponding integrable bi-Hamiltonian hierarchies were constructed by the present author in the papers [4], [5], and for compatible nonlocal Mokhov–Ferapontov brackets [6] generated by metrics of constant Riemannian curvature, the bi-Hamiltonian hierarchies were constructed in [7].