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.