Non-integrability of the Sasano system of type $A^{(2)}_5$

TL;DR

This paper proves that the Sasano system of type A^{(2)}_5, a complex four-dimensional Hamiltonian system with affine Weyl group symmetry, is non-integrable by rational first integrals for all parameter values admitting rational solutions.

Contribution

It applies Morales-Ramis-Simó theory to rigorously establish non-integrability of the Sasano system of type A^{(2)}_5 for all relevant parameters.

Findings

Proves non-integrability for all parameter values with rational solutions

Utilizes Morales-Ramis-Simó theory for Hamiltonian systems

Confirms non-integrability of the specific Sasano system

Abstract

The Sasano sytem of type $A^{(2)}_5$ is a four-dimensional non-linear system of ordinary differential equations, which has an affine Weyl group of symmetries of type $A^{(2)}_5$. It is also a tipe dependent Hamiltonian system, which can be considered as coupled Painlev\'e III systems. In this paper, utilizing the Morales-Ramis-Sim\'o theory for integrability of Hamiltonian systems, we prove rigorously that for all values of the parameters, for which the Sasano system of type $A^{(2)}_5$ admits a particular rational solution, it is non-integrable by rational first integrals.