Are all classical superintegrable systems in two-dimensional space linearizable?

Several examples of classical superintegrable systems in a two-dimensional space are shown to possess hidden symmetries leading to their linearization. They include those determined fifty years ago in the work of Fris et al. [Phys. Lett. 13, 354–356 (1965)], their generalizations, and the more recent Tremblay-Turbiner-Winternitz system [F. Tremblay et al., J. Phys. A: Math. Theor. 42, 242001 (2009)]. We conjecture that all classical superintegrable systems in the two-dimensional space have hidden symmetries that make them linearizable.