  Liouville (super)integrability of a Hamiltonian system of differential
equations is based on the existence of globally well-defined constants of the
motion, while Lie point symmetries provide a local approach to conserved
integrals. Therefore, it seems natural to investigate in which sense Lie point
symmetries can be used to provide information concerning the superintegrability
of a given Hamiltonian system. The two-dimensional oscillator and the central
force problem are used as benchmark examples to show that the relationship
between standard Lie point symmetries and superintegrability is neither
straightforward nor universal. In general, it turns out that superintegrability
is not related to either the size or the structure of the algebra of
variational dynamical symmetries. Nevertheless, all of the first integrals for
a given Hamiltonian system can be obtained through an extension of the standard
point symmetry method, which is applied to a superintegrable nonlinear
oscillator describing the motion of a particle on a space with non-constant
curvature and spherical symmetry.
