  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.
