Consistency of the B\"acklund transformation for the spin Calogero-Moser system

TL;DR

This paper proves the consistency of the Bäcklund transformation for the spin Calogero-Moser system across rational, trigonometric, and hyperbolic cases, ensuring the transformation's well-definedness and uniqueness.

Contribution

It establishes the consistency of the Bäcklund transformation for the spin Calogero-Moser system in multiple cases by analyzing associated functions and proving their identically zero property.

Findings

Bäcklund transformation is consistent for sCM system in various cases.

Constructed functions measuring overdetermined system's departure are identically zero.

Ensured unique solutions to the initial value problem for the BT.

Abstract

We prove the consistency of the B\"{a}cklund transformation (BT) for the spin Calogero-Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ODEs; to establish our result, we construct and analyze certain functions that measure the departure of this overdetermined system from consistency and show, under mild assumptions, that these functions are identically zero and that this allows for a unique solution to the initial value problem for the overdetermined system.