On integrability of the equations for nonsingular pairs of compatible flat metrics

In this paper, we deal with the problem of description of nonsingular pairs of compatible flat metrics for the general N -component case. We describe the scheme of the integrating the nonlinear equations describing nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics). This scheme was announced in our previous paper [1]. It is based on the reducing this problem to a special reduction of the Lamé equations and the using the Zakharov method of differential reductions [2] in the dressing method (a version of the inverse scattering method). We shall use both contravariant metrics g(u) with upper indices, where u = (u, ..., u) are local coordinates, 1 ≤ i, j ≤ N , and covariant metrics gij(u) with lower indices, g (u)gsj(u) = δ j . The indices of coefficients of the Levi–Civita connections Γ i jk(u) (the Riemannian connections generated by the corresponding metrics) and tensors of Riemannian curvature R jkl(u) are raised and lowered by the metrics corresponding to them: