A universal method to approach the Poincar\'e center problem

We address the classical (degenerate or non-degenerate) center problem posed by Poincar\'e in the 19th century for monodromic singularities of analytic families of planar vector fields $\mathcal{X}$. We prove that every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\mathcal{X}$ has no local curves of zero angular speed, the Poincar\'e map is analytic, and if, in addition, $V$ has an essential singularity, then the singularity of $\mathcal{X}$ is a center. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize any center of a polynomial vector field. Applications to nontrivial families that have resisted other methods are also provided.