The center problem for Z_2-symmetric nilpotent vector fields
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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.