LAUSR

Abstract We say that a polynomial differential system x ˙ = P ( x , y ) , y ˙ = Q ( x , y ) having the origin as a singular point is Z 2 -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 Z 2 -symmetric system 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.