  We present a method to obtain higher order integrals and polynomial algebras
for two-dimensional superintegrable systems from creation and annihilation
operators. All potentials with a second and a third order integrals of motion
separable in Cartesian coordinates were studied. The integrals of motion of two
of them do not generate a cubic algebra. We construct for these Hamiltonians a
higher order polynomial algebra from the creation and annihilation operators.
We obtain quintic and seventh order polynomial algebras. We give also for
polynomial algebras of order 7 realizations in terms of deformed oscillator
algebras. These realizations and finite dimensional unitary representations
allow us to obtain the energy spectrum.
