Lax pairs for the equations describing compatible nonlocal Poisson brackets of hydrodynamic type, and integrable reductions of the Lamé equations

In the present work, the nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demonstrate the integrability of the equations for some especially important partial classes of compatible nonlocal Poisson brackets of hydrodynamic type, in particular, for the most important case when one of the compatible Poisson brackets is local and also for the case when one of the compatible Poisson brackets is generated by a metric of constant Riemannian curvature. The case when one of the compatible Poisson brackets of hydrodynamic type is local and nondegenerate was studied in detail in our previous paper [1], where the corresponding equations were derived and the integrability of these equations was announced. This case is very important, since, as was shown in [1], any solution of these equations generates an integrable bi-Hamiltonian hierarchy of hydrodynamic type by explicit formulae. Moreover, these equations describe an important class of integrable reductions of the classical Lamé equations. Accordingly, in the case when one of the compatible Poisson brackets is generated by a metric of constant Riemannian curvature, the corresponding equations describe integrable reductions of the equations for orthogonal curvilinear coordinate systems in spaces of constant curvature.