Existence of inverse integrating factors and Lie symmetries for degenerate planar centers

We prove that any (possibly degenerate) center of an analytic planar system admits a C∞ inverse integrating factor and a C∞ Lie symmetry in a neighborhood of the singularity. We also show that there exists an analytic degenerate center for which any C∞ inverse integrating factor is flat at the singular point, and that a sufficient condition for the existence of an analytic inverse integrating factor is the existence of an analytic first integral.