Center problem for generic degenerate vector fields

We generalize the method of construction of an integrating factor for Abel differential equations, developed in Briskin et al. (1998), for any generic monodromic singularity. Here generic means that the vector field has not characteristic directions in the quasi-homogeneous leading term in certain coordinates. We apply this method to some degenerate differential systems

On the center conditions for analytic monodromic degenerate singularities

In this paper we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields can be applied independently that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allows to study monodromic degenerate singularities with characteristic directions