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We consider classical and quantum one and two-dimensional systems with ladderoperators that satisfy generalized Heisenberg algebras. In the classical case,this construction is related to the existence of closed trajectories. Inparticular, we apply these results to the infinite well and Morse potentials.We discuss how the degeneracies of the permutation symmetry of quantumtwo-dimensional systems can be explained using products of ladder operators.These products satisfy interesting commutation relations. The two-dimensionalMorse quantum system is also related to a generalized two-dimensional Morsesupersymmetric model. Arithmetical or accidental degeneracies of such systemare shown to be associated to additional supersymmetry.

Generalized Heisenberg Algebras, SUSYQM and Degeneracies: Infinite Well and Morse Potential