The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C∞C∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C∞C∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.