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

On realizations of polynomial algebras with three generators via deformed oscillator algebras