Analytic normalization of analytically integrable differential systems near a periodic orbit

Abstract

For an analytic differential system in $\mathbb R^n$ with a periodic orbit, we will prove that if the system is analytically integrable around the periodic orbit, i.e. it has $n-1$ functionally independent analytic first integrals defined in a neighborhood of the periodic orbit, then the system is analytically equivalent to its Poincar\'e--Dulac type normal form. This result is an extension for analytic integrable differential systems around a singularity to the ones around a periodic orbit.