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${{\mathbb{R}}^{5}}$
We study Dupin hypersurfaces in ${{\mathbb{R}}^{5}}$ parametrized by lines of curvature, with four distinct principal curvatures. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We show that these vector valued functions are invariant by inversions and homotheties.
