Mokhov, O. I. Integrability of the equations for nonsingular pairs of compatible flat metrics. (English. Russian original) Zbl 1029.37046 Theor. Math. Phys. 130, No. 2, 198-212 (2002); translation from Teor. Mat. Fiz. 130, No. 2, 233-250 (2002). Summary: We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general \(N\)-component case. This problem is equivalent to the problem of describing all compatible Dubrovin-Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lamé equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lamé equations (a version of the inverse scattering method). Cited in 4 Documents MSC: 37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Keywords:nonsingular flat pencils of metrics; Dubrovin-Novikov brackets; integrable systems of hydrodynamic type; differential reduction; Lamé equations; inverse scattering; Zakharov method PDFBibTeX XMLCite \textit{O. I. Mokhov}, Theor. Math. Phys. 130, No. 2, 198--212 (2002; Zbl 1029.37046); translation from Teor. Mat. Fiz. 130, No. 2, 233--250 (2002) Full Text: DOI arXiv