  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 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 quantum
reducible and irreducible rational potentials separable in Cartesian
coordinates in E2. We also discuss these potentials from the point of view of
supersymmetric and PT-symmetric quantum mechanics.
