Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs

TL;DR

This paper introduces LSSS, a computer algebra program optimized for solving large sparse linear systems, and applies it to analyze symmetries of a non-abelian Laurent ODE, confirming the completeness of a known Lax pair.

Contribution

The paper presents a new efficient solver for large sparse linear systems and demonstrates its application in symmetry analysis of a non-abelian Laurent ODE.

Findings

LSSS effectively solves large sparse systems with many zero variables.

Symmetry computations confirmed the Lax pair generates all integrals up to degree 14.

The approach advances computational methods in integrable non-abelian ODEs.

Abstract

The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M. Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14.