Generalized Deformed su(2) Algebras, Deformed Parafermionic Oscillators and Finite W Algebras

Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to posses the structure of a generalized deformed su(2) algebra, the representation theory of which is known. Furthermore, the generalized deformed parafermionic oscillator is identified with the algebra of several physical systems (isotropic oscillator and Kepler system in 2-dim curved space, Fokas--Lagerstrom, Smorodinsky--Winternitz and Holt potentials) and mathematical constructions (generalized deformed su(2) algebra, finite W algebras $\bar {\rm W}_0$ and W$_3^{(2)}$). The fact that the Holt potential is characterized by the W$_3^{(2)}$ symmetry is obtained as a by-product.