Symmetries and integrals of motion of a superintegrable deformed oscillator

Highlights: • Superintegrable deformation of harmonic oscillator is considered. • The algebra of symmetry generators is analyzed in detail. • The explicit solutions to the equations of motion are found. • The general construction of superintegrable systems is sketched. The symmetry structure of twodimensional nonlinear isotropic oscillator, introduced in Ballesteros et al. (2008), is discussed. It is shown that it possesses three independent integrals of motion which can be chosen in such a way that they span $SU\left(2\right)$, $E\left(2\right)$ or $SU(1,1)$ algebras, depending on the value of total energy. They generate the infinitesimal canonical symmetry transformations; integrability of the latter is analyzed. The results are then generalized to the case of arbitrary number of degrees of freedom.