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