Galoisian obstructions to non-Hamiltonian integrability

TL;DR

This paper extends the Morales--Ramis--Simo theorem on Galoisian obstructions from Hamiltonian to non-Hamiltonian systems, showing that integrability implies virtually Abelian Galois groups of variational equations.

Contribution

It generalizes the Galoisian obstruction criterion for integrability to non-Hamiltonian dynamical systems, broadening the scope of the original theorem.

Findings

Non-Hamiltonian integrability implies virtually Abelian Galois groups.

The extension applies to variational equations of any order.

Provides a new tool for analyzing integrability in broader dynamical systems.

Abstract

We show that the main theorem of Morales--Ramis--Simo about Galoisian obstructions to meromorphic integrability of Hamiltonian systems can be naturally extended to the non-Hamiltonian case. Namely, if a dynamical system is meromorphically integrable in the non-Hamiltonian sense, then the differential Galois groups of the variational equations (of any order) along its solutions must be virtually Abelian