On Integrability of the Equations for Nonsingular Pairs of Compatible   Flat Metrics
  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). It is based on the reducing this problem to a special reduction of the Lame equations and the using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method). 
O. I. Mokhov