A nonclassical algebraic solution of a 3-variable irregular Garnier system

TL;DR

This paper constructs a new nonclassical algebraic solution for a 3-variable irregular Garnier system, expanding the known solutions beyond the classical and pull-back types.

Contribution

It provides the explicit form of the remaining algebraic solution for the 3-variable irregular Garnier system, complementing previous classifications.

Findings

Explicit form of the new algebraic solution is derived.

Completes the classification of nonclassical solutions for 3-variable systems.

Enhances understanding of algebraic solutions in irregular Garnier systems.

Abstract

In this paper, a nonclassical algebraic solution of a 3-variable irregular Garnier system is constructed. Diarra--Loray have studied classification of algebraic solutions of irregular Garnier systems. There are two type of the algebraic solutions: classical type and pull-back type. They have shown that there are exactly three nonclassical algebraic solutions for $N$-variables irregular Garnier systems with $N>1$. Explicit forms of two of the three solutions are already given. The solution constructed in the present paper is the remained algebraic solution.