Master symmetries, non-Hamiltonian symmetries and superintegrability of the generalized Smorodinsky-Winternitz system

The first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach. It is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries. In the second part, the properties of dynamical but non-Hamiltonian symmetries are studied. It is proved that the higher order superintegrability of the generalized Smorodinsky-Winternitz system is a consequence of the existence of symplectic symmetries not preserving the Hamiltonian function.