On the Poincaré-Lyapunov centre theorem (Q1577426)

The classical Poincaré-Lyapunov centre theorem says that a centre-type singularity of an analytic vector field on the plane has an analytic first integral, provided that the linear part of the vector field at the singular point generates a nontrivial rotation. The authors consider the multidimensional analog of the planar centre dynamics, and give a geometric proof and a slight generalization of a result of \textit{M.~Urabe} and \textit{Y.~Sibuya} [J. Sci. Hiroshima Univ., Ser. A 19, 87-100 (1955; Zbl 0073.31001)], stating that an analytic multicentre, with linear part at the singular point which generates a multirotation, is always linearizable. The main tools in the proof are the analytic extension up to the origin of the period function, obtained through the analysis of the complexification of the vector field, and the use of the Bochner theorem on the linearization of compact group actions in a neighbourhood of a fixed point. As a byproduct of these arguments, a still another proof of the Poincaré-Lyapunov centre theorem is obtained.