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

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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 W₀) 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 W₀ and W₃⁽²⁾). The fact that the Holt potential is characterized by the W₃⁽²⁾ symmetry is obtained as a by-product.