We say that a polynomial differential system ˙x = P(x, y), ˙y = Q(x, y) having the origin as a singular point is Z2-symmetric if P(−x, −y) = −P(x, y) and Q(−x, −y) = −Q(x, y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C∞ first integral. But up to know there are no characterized these two kinks of nilpotent centers. Here we prove that the origin of any Z2-symmetric is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x, y) = y 2 + · · ·, where the dots denote terms of degree higher than two.

The first and second authors are partially supported by a MINECO/FEDER grant number MTM2014-56272-C2-2 and by the Consejer´ıa de Educaci´on y Ciencia de la Junta de Andaluc´ıa (projects P12-FQM-1658, FQM-276). The third author is partially supported by a MINECO/FEDER grant number MTM2017-84383-P and by an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The fourth author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568.