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

Abstract

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.