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We consider a superintegrable Hamiltonian system in a two-dimensional spacewith a scalar potential that allows one quadratic and one cubic integral ofmotion. We construct the most general associative cubic algebra and we presentspecific realizations. We use them to calculate the energy spectrum. Allclassical and quantum superintegrable potentials separable in cartesiancoordinates with a third order integral are known. The general formalism isapplied to one of the quantum potentials

Erratum: Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion [J. Math. Phys. 48, 012902 (2007)]