Normalization of rationally integrable systems

TL;DR

This paper extends the analytic normalization results for integrable systems to those with rational first integrals and commuting vector fields, broadening the class of systems where normalization is possible.

Contribution

It introduces a method to achieve local analytic normalization for rationally integrable systems, expanding previous results from purely analytic to rational cases.

Findings

Normalization achieved for rationally integrable systems.

Broadens applicability of normalization techniques.

Provides a framework for systems with rational first integrals.

Abstract

In two previous papers we showed that any analytically integrable vector field admits a local analytic Poincar\'e-Birkhoff normalization in the neighborhood of a singular point. The aim of this paper is to extend this analytic normalization result to the case of rationally integrable systems, where the first integrals and commuting vector fields are not required to be analytic, but just rational (i.e., quotients of analytic functions or vector fields by analytic functions).