# Non-integrability of the Painlev\'{e} IV equation in the Liouville-Arnold sense and Stokes phenomena

## TL;DR

This paper proves the non-integrability of a Hamiltonian system related to the Painlevé IV equation using Galois theory and Stokes phenomena, showing it cannot be solved by meromorphic first integrals.

## Contribution

It demonstrates the non-integrability of the Painlevé IV Hamiltonian system in the Liouville-Arnold sense through explicit Galois group analysis and Stokes matrices, extending previous results.

## Key findings

- The Hamiltonian system is not integrable in the Liouville-Arnold sense.
- The Galois group associated with the second variational equations is non-Abelian.
- The non-integrability holds for all parameter values where Painlevé IV has rational solutions.

## Abstract

In this paper we study the integrability of the Hamiltonian system associated to the fourth Painlev\'{e} equation. We prove that one two parametric family of this Hamiltonian system is not integrable in the sense of the Liouville-Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations we deduce that the connected component of the unit element of the corresponding Galois grou is not Abelian. Thus the Morales-Ramis-Sim\'{o} theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for allvalues of the parameters for which the $P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in $t$.