  We present the most general polynomial Lie algebra generated by a second
order integral of motion and one of order M, construct the Casimir operator,
and show how the Jacobi identity provides the existence of a realization in
terms of deformed oscillator algebra. We also present the classical analog of
this construction for the most general Polynomial Poisson algebra. Two specific
classes of such polynomial algebras are discussed that include the symmetry
algebras observed for various 2D superintegrable systems.
