A characterization of centres via first integrals (Q1117377)

An isolated critical point of a smooth differential system \(x'=f(x)\) in the plane is proved to be a centre iff there exists a local smooth first integral which has the point in question as an isolated minimum and also iff there exists a local invariant measure which is equivalent with the Lebesgue measure and has smooth density. The same type of characterization of the centres at infinity by first integrals is also proved.