Liouville canonical form for compatible nonlocal Poisson brackets of hydrodynamic type, and integrable hierarchies

In the present paper, we solve the problem of reducing to the simplest and convenient for our purposes, “canonical” form for an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets [1]) in order to get an effective construction of the integrable hierarchies related to all these compatible Poisson brackets. As was shown in [2], [3] (see also [4]), compatible Mokhov–Ferapontov brackets are described by a consistent nonlinear system of equations integrable by the method of inverse scattering problem (the case of flat metrics see in [5]–[7]). But the problem of an effective construction of the corresponding integrable hierarchies in this case, what is the main purpose of the present paper, requires a different approach to the description of these compatible brackets. In this paper, for an arbitrary solution of the integrable system of equations describing the compatible brackets under consideration, that is, for an arbitrary pair of these compatible brackets, integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible nonlocal Poisson brackets of hydrodynamic type are constructed in an explicit form. For the case of the Dubrovin–Novikov brackets [8] (the local Poisson brackets of hydrodynamic type), this problem was considered and completely solved in the present author’s works [9], [10]. In [1] the nonlocal Poisson brackets of hydrodynamic type which have the following form (the Mokhov–Ferapontov brackets):