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Using the idea of shape invariance with respect to the main
quantum number n, we construct here new generators for the
realization of commutative relations of Lie algebra
\emph{gl(2,c)}. By an appropriate parametrization, new quantum
solvable Hamiltonians with dynamical symmetry group \emph{GL(2,c)}
and infinite-fold degeneracy are derived. These models correspond
to the motion of a free particle on \emph{SL(2,c)}/\emph{GL(1,c)}.
It is shown that the related quantum states satisfy shape
invariance equations with respect to n, and they also represent
the Lie algebra \emph{gl(2,c)}.
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