  The integrals of motion of the classical two dimensional superintegrable
systems with quadratic integrals of motion close in a restrained quadratic
Poisson algebra, whose the general form is investigated. Each classical
superintegrable problem has a quantum counterpart, a quantum superintegrable
system. The quadratic Poisson algebra is deformed to a quantum associative
algebra, the finite dimensional representations of this algebra are calculated
by using a deformed parafermion oscillator technique. It is shown that, the
finite dimensional representations of the quadratic algebra are determined by
the energy eigenvalues of the superintegrable system. The calculation of energy
eigenvalues is reduced to the solution of algebraic equations, which are
universal for all two dimensional superintegrable systems with quadratic
integrals of motion.
