## Bäcklund transformations for many-body systems related to KdV.(English)Zbl 0952.37054

The authors present some new examples of Bäcklund transformations for $$n$$-body systems, namely the many-body generalization of the integrable Hénon-Heiles system, the Garnier system and the Neumann system on the sphere. Bäcklund transformations for these systems are obtained by reduction of the standard Bäcklund transformation for KdV, which arises from the Darboux-Crum transformation for Schrödinger operators. The restriction of the Darboux transformation to the stationary flows of the modified KdV hierarchy has been discussed. The Bäcklund transformation for the $$sl(2)$$ Gaudin magnet with quasiperiodic boundary condition is given.

### MSC:

 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 39A12 Discrete version of topics in analysis
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