  We solve the problem of description for nonsingular pairs of compatible flat
metrics in the general N-component case. The integrable nonlinear partial
differential equations describing all nonsingular pairs of compatible flat
metrics (or, in other words, nonsingular flat pencils of metrics) are found and
integrated. The integrating of these equations is based on reducing to a
special nonlinear differential reduction of the Lame equations and using the
Zakharov method of differential reductions in the dressing method (a version of
the inverse scattering method).
