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We consider a superintegrable quantum potential in two-dimensional Euclideanspace with a second and a third order integral of motion. The potential iswritten in terms of the fourth Painleve transcendent. We construct for thissystem a cubic algebra of integrals of motion. The algebra is realized in termsof parafermionic operators and we present Fock type representations which yieldthe corresponding energy spectra. We also discuss this potential from the pointof view of higher order supersymmetric quantum mechanics and obtain groundstate wave functions.

Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve transcendent potentials