Classification of Hamiltonian non-abelian Painlev\'e type systems

TL;DR

This paper classifies all Hamiltonian non-abelian Painlevé systems of types P1 to P6 with constant coefficients, introducing non-abelian constants and establishing integrability via isomonodromic Lax pairs.

Contribution

It provides a complete classification of Hamiltonian non-abelian Painlevé systems and proves integrability of new systems through Lax pairs.

Findings

All Hamiltonian non-abelian Painlevé systems of types P1-P6 identified.

New non-abelian constants introduced in P1-P5 systems.

Integrability of new P3' and P5 systems demonstrated via Lax pairs.

Abstract

All Hamiltonian non-abelian Painlev\'e systems of ${\rm{P}}_{1}-{\rm{P}}_{6}$ type with constant coefficients are found. For ${\rm{P}}_{1}-{\rm{P}}_{5}$ systems, we replace an appropriate inessential constant parameter with a non-abelian constant. To prove the integrability of new ${\rm{P}}_{3}^{\prime}$ and ${\rm{P}}_{5}$ systems thus obtained, we find isomonodromic Lax pairs for them.