A note on local integrability of differential systems

For an n-dimensional local analytic differential system 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 whether the system has an analytic first integral at the origin provided that the origin is not an isolated singular point remains open.