  We introduce the most general quartic Poisson algebra generated by a second
and a fourth order integral of motion of a 2D superintegrable classical system.
We obtain the corresponding quartic (associative) algebra for the quantum
analog and we extend Daskaloyannis' construction in obtained in context of
quadratic algebras and we obtain the realizations as deformed oscillator
algebras for this quartic algebra. We obtain the Casimir operator and discuss
how these realizations allow to obtain the finite dimensional unitary
irreductible representations of quartic algebras and obtain algebraically the
degenerate energy spectrum of superintegrable systems. We apply the
construction and the formula obtained for the structure function on a
superintegrable system related to type I Laguerre exceptionnal orthogonal
polynomials introduced recently.
