We solve the problem of description of nonsingular pairs of compatible flat metrics for the general -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 Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).
Oleg I. Mokhov. "Compatible flat metrics." J. Appl. Math. 2 (7) 337 - 370, 24 November 2002. https://doi.org/10.1155/S1110757X02203149