LIE SYMMETRIES AND THE CENTER PROBLEM

In this short survey we study the narrow relation between the center problem and the Lie symmetries. It is well known that an analytic vector eld X having a non-degenerate center has a non-trivial analytic Lie symmetry in a neighborhood of it, i.e. there exists an analytic vector eld Y such that [X;Y] = \(\mu\)X. The same happens for a nilpotent center with an analytic rst integral as can be seen from the last results about nilpotent centers. From the last results for nilpotent and degenerate centers it also can be proved that any nilpotent or degenerate center has a trivial smooth (of class \(C^{\infty} \) ) ) Lie symmetry. Remains open if always exists also a non-trivial Lie symmetry for any nilpotent and degenerate center.