Bäcklund transformations for many-body systems related to KdV

We present Bäcklund transformations (BTs) with parameter for certain classical integrable n -body systems, namely the many-body generalized Hénon-Heiles, Garnier and Neumann systems. Our construction makes use of the fact that all these systems may be obtained as particular reductions (stationary or restricted flows) of the KdV hierarchy; alternatively they may be considered as examples of the reduced sl (2) Gaudin magnet. The BTs provide exact time-discretizations of the original (continuous) systems, preserving the Lax matrix and hence all integrals of motion, and satisfy the spectrality property with respect to the Bäcklund parameter.