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Journal of Applied Mathematics
Volume 2 (2002), Issue 7, Pages 337-370

Compatible flat metrics

Centre for Nonlinear Studies, L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 2 Kosygina Street, Moscow 117940, Russia

Received 13 December 2001

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We solve the problem of description of nonsingular pairs of compatible flat metrics for 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 Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).