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

摘要

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.