A note on local integrability of differential systems

For an $n$–dimensional local analytic differential system $\dot{x}=Ax+f(x)$ with $f(x)=O(\mathrm{\mid }x{\mathrm{\mid }}^2)$, 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.