Symmetries and convergence of normalizing transformations (Q1329289)

Normal forms of 2-dimensional systems of autonomous ordinary differential equations are considered in a neighbourhood of a stationary point. It is supposed that the linear part is semisimple, non-zero and the normal form contains infinitely many terms. The author proves the following theorem: The system (under the above conditions) allows a convergent transformation into normal form if and only if it admits a local one- parameter group of symmetries. The paper contains a general discussion, conjectures on existence of convergent normalizing transformations.