  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.
