Characterization of centers by their complex separatrices

In this work, we address analytic families of real planar vector fields having a monodromic singularity at the origin for all parameters, where Λ is an open subset of the real finite-dimensional Euclidean space. We assume that depends analytically on λ. This naturally leads to the so-called center-focus problem, which consists of describing the partition of Λ induced by the centers and the foci at the origin. We provide a characterization of the centers (whether degenerate or not) in terms of a specific integral of the cofactor associated with a real invariant analytic curve passing through the singularity, which always exists. Several consequences and applications are also discussed.