An elliptic Garnier system

TL;DR

This paper introduces an elliptic analogue of the Garnier system through a linear difference equation system expressed with theta functions, providing a Lax pair for the elliptic Painlevé equation.

Contribution

It constructs a new elliptic Garnier system with a Lax pair, linking it explicitly to the elliptic Painlevé equation and expanding the understanding of elliptic isomonodromic deformations.

Findings

Identified the elliptic Garnier system as an analogue of classical systems.

Provided explicit Lax pair for the elliptic Painlevé equation.

Described the system's singularity structure and parameterization.

Abstract

We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at $4m+12$ points for $m \geq 1$, which appear in pairs due to a symmetry condition. We parameterize this linear system in terms a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which $m=1$ with the elliptic Painlev\'e equation, hence, this work provides an explicit form and Lax pair for the elliptic Painlev\'e equation.