A note on local integrability of differential systems

Abstract

For an $n$--dimensional local analytic differential system $\dot{x}=Ax+f(x)$ with $f(x)=O(|x{|}^2)$, the Poincaré 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.

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References in corpus (2)

• Analytic integrable systems: Analytic normalization and embedding flows
• Generalized rational first integrals of analytic differential systems