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Superintegrable systems in 2D Darboux spaces were classified and it was foundthat there exist 12 distinct classes of superintegrable systems with quadraticintegrals of motion (and quadratic symmetry algebras generated by theintegrals) in the Darboux spaces. In this paper, we obtain exact solutions viapurely algebraic means for the energies of all the 12 existing classes ofsuperintegrable systems in four different 2D Darboux spaces. This is achievedby constructing the deformed oscillator realization and finite-dimensionalirreducible representation of the underlying quadratic symmetry algebragenerated by quadratic integrals respectively for each of the 12superintegrable systems. We also introduce generic cubic and quintic algebras,generated respectively by linear and quadratic integrals and linear and cubicintegrals, and obtain their Casimir operators and deformed oscillatorrealizations. As examples of applications, we present three classes of newsuperintegrable systems with cubic symmetry algebras in 2D Darboux spaces.

Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces