The center problem for Z2-symmetric nilpotent vector fields

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. However these two kinds of nilpotent centers are not characterized for different families of differential systems. Here we prove that the origin of any Z2-symmetric system is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x,y)=y2+⋯, where the dots denote terms of degree higher than two.