Elliptic Linear Problem for Calogero-Inozemtsev Model and Painleve VI Equation

TL;DR

This paper constructs a Lax pair for the Calogero-Inozemtsev model, deriving a $2\times 2$ representation for the one-degree-of-freedom case, and connects it to the elliptic form of Painleve VI, demonstrating algebraic integrability.

Contribution

It introduces a new Lax pair for the Calogero-Inozemtsev model and links it to the elliptic Painleve VI equation, advancing understanding of integrable systems.

Findings

Derived a $3N\times 3N$ Lax pair with spectral parameter.

Reduced to a $2\times 2$ Lax representation for one degree of freedom.

Proved algebraic integrability of the model.

Abstract

We introduce $3N\times 3N$ Lax pair with spectral parameter for Calogero-Inozemtsev model. The one degree of freedom case appears to have $2\times 2$ Lax representation. We derive it from the elliptic Gaudin model via some reduction procedure and prove algebraic integrability. This Lax pair provides elliptic linear problem for the Painlev{\'e} VI equation in elliptic form.