  We consider a superintegrable Hamiltonian system in a two-dimensional space
with a scalar potential that allows one quadratic and one cubic integral of
motion. We construct the most general associative cubic algebra and we present
specific realizations. We use them to calculate the energy spectrum. All
classical and quantum superintegrable potentials separable in cartesian
coordinates with a third order integral are known. The general formalism is
applied to one of the quantum potentials.
