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A nonstandard realization of the su(1,1) algebra is used to extract a two-parameter class of scattering potentials as well as to calculate the reflection coefficient of the associated one-dimensional scattering problem in the spirit of the algebraic scattering theory. The nontrivial geometric content of such realizations is discussed, and an interesting connection with geometric phases is pointed out. It is argued that using larger noncompact groups, realizations related to non-Abelian geometric phases may be useful for obtaining analytical expressions for interaction terms corresponding to higher-dimensional scattering problems.