A note on local integrability of differential systems

TL;DR

This paper discusses conditions under which local analytic differential systems have first integrals, focusing on the unresolved question of whether non-isolated singular points guarantee an analytic first integral.

Contribution

It clarifies the open problem regarding the existence of analytic first integrals at non-isolated singular points in differential systems.

Findings

Extends Poincaré's nonintegrability theorem to certain cases.

Highlights unresolved question about non-isolated singular points.

Provides conditions for existence of formal first integrals.

Abstract

For an $n$--dimensional local analytic differential system $\dot x=Ax+f(x)$ with $f(x)=O(|x|^2)$, the Poincar\'e nonintegrability theorem states that if the eigenvalues of $A$ are not resonant, the system does not have an analytic or a formal first integral in a neighborhood of the origin. This result was extended in 2003 to the case when $A$ admits one zero eigenvalue and the other are non--resonant: for $n=2$ the system has an analytic first integral at the origin if and only if the origin is a non--isolated singular point; for $n>2$ the system has a formal first integral at the origin if and only if the origin is not an isolated singular point. However, the question of \emph{whether the system has an analytic first integral at the origin provided that the origin is not an isolated singular point} remains open.