The center problem for Z<inf>2</inf>-symmetric nilpotent vector fields

Antonio Algaba, Cristóbal García, Jaume Giné, Jaume Llibre

Research output: Contribution to journalArticleResearch

10 Citations (Scopus)


© 2018 Elsevier Inc. 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.
Original languageEnglish
Pages (from-to)183-198
JournalJournal of Mathematical Analysis and Applications
Publication statusPublished - 1 Oct 2018


  • Center problem
  • Nilpotent singularity
  • Z -symmetric differential systems 2


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