On the centers of planar analytic differential systems

In this work we continue the study of the centers which are limits of linear type centers. It is proved that if a degenerate center has an inverse integrating factor V (x, y) with V (0, 0) not equal 0, then this degenerate center is also the limit of linear type centers. Moreover, we show that the degenerate centers with characteristic directions that are the limits of degenerate centers without characteristic directions are also detectable, from a theoretical point of view, with the Bautin method.