Using two different types of the laddering equations realized simultaneously by the associated Gegenbauer functions, we show that all quantum states corresponding to the motion of a free particle on AdS2 and S2 are splitted into infinite direct sums of infiniteand finite-dimensional Hilbert subspaces which represent Lie algebras u(1, 1) and u(2) with infinite- and finite-fold degeneracies, respectively. In addition, it is shown that the representation bases of Lie algebras with rank 1, i.e., gl(2, C), realize the representation of nonunitary parasupersymmetry algebra of arbitrary order. The realization of the representation of parasupersymmetry algebra by the Hilbert subspaces which describe the motion of a free particle on AdS2 and S2 with the dynamical symmetry groups U(1, 1) and U(2) are concluded as well.
