We present a description of isochronous centres of planar vector fields by means of their groups of symmetries. More precisely, given a normalizer of (i.e., , where is a scalar function), we provide a necessary and sufficient isochronicity condition based on . This criterion extends the result of Sabatini and Villarini that establishes the equivalence between isochronicity and the existence of commutators (). We put also special emphasis on the mechanical aspects of isochronicity; this point of view forces a deeper insight into the potential and quadratic-like Hamiltonian systems. For these families we provide new ways to find isochronous centres, alternative to those already known from the literature.
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Emilio Freire, Armengol Gasull, Antoni Guillamon, A characterization of isochronous centres in terms of symmetries. Rev. Mat. Iberoam. 20 (2004), no. 1, pp. 205–222DOI 10.4171/RMI/386