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SO(2,1) is the symmetry algebra for a class of three-parameter problems thatincludes the oscillator, Coulomb and Morse potentials as well as other problemsat zero energy. All of the potentials in this class can be mapped into theoscillator potential by point canonical transformations. We call this class the"oscillator class". A nontrivial graded extension of SO(2,1) is defined and itsrealization by two-dimensional matrices of differential operators acting inspinor space is given. It turns out that this graded algebra is thesupersymmetry algebra for a class of relativistic potentials that includes theDirac-Oscillator, Dirac-Coulomb and Dirac-Morse potentials. This class is, infact, the relativistic extension of the oscillator class. A new point canonicaltransformation, which is compatible with the relativistic problem, isformulated. It maps all of these relativistic potentials into theDirac-Oscillator potential.Comment: Replaced with a more potrable PDF versio

Graded extension of so(2,1) Lie algebra and the search for exact solutions of the Dirac equation by point canonical transformations