Published in Respondek, W., & Tall, I. A. (2001). How many symmetries does admit a nonlinear single-input control system around an equilibrium? Proceedings of the IEEE Conference on Decision and Control, v 2, 1795-1800. doi: 10.1109/.2001.981165. ©2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.


We describe all symmetries of a single-input nonlinear control system, that is not feedback linearizable and whose first order approximation is controllable, around an equilibrium point. For a system such that a feedback transformation, bringing it to the canonical form, is analytic we prove that the set of all local symmetries of the system is exhausted by exactly two 1-parameter families of symmetries, if the system is odd, and by exactly one 1-parameter family otherwise. We also prove that the form of the set of symmetries is completely described by the canonical form of the system: possessing a nonstationary symmetry, a 1-parameter family of symmetries, or being odd corresponds, respectively, to the fact that the drift vector field of the canonical form is periodic, does not depend on the first variable, or is odd. If the feedback transformation bringing the system to its canonical form is formal, we show an analogous result for an infinitesimal symmetry: its existence is equivalent to the fact that the drift vector field of the formal canonical form does not depend on the first variable. We illustrate our results by studying symmetries of the variable length pendulum.