Nonlinearizable single-input control systems do not admit stationary symmetries

The main result of the paper states that almost any analytic single-input control system, which is truly nonlinear, that is not feedback linearizable, with controllable linearization at an equilibrium point, does not admit any symmetry preserving that point. By almost any system, we mean that we exclude a small class of odd systems, that admit just one nontrivial symmetry conjugated to minus identity. The obtained results are based on a recent classification of nonlinear single-input systems under formal feedback. We also describe symmetries of feedback linearizable systems. (C) 2002 Elsevier Science B.V. All rights reserved.