The description of pairs of compatible first-order differential geometric poisson brackets

We show that bi-Hamiltonian structures of systems of hydrodynamic type can be described in terms of solutions of nonlinear equations. These equations can be integrated by the inverse scattering transform for arbitrary metrics as well as for flat ones. In particular, if one metric is flat and the other has a constant curvature, then the corresponding integrable system is a reduction of the Cherednik chiral-field model.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 293–309, February, 2005.