Consider a hypersurface $M^n$ in $mathbb{R}^{n+1}$ with $n$ distinct principal curvatures, parametrized by lines of curvature with vanishing Laplace invariants. (1) If the lines of curvature are planar, then there are no such hypersurfaces for $n{geq}4$, and for $n=3$, they are, up to M$ddot{o}$bius transformations Dupin hypersurfaces with constant M$ddot{o}$bius curvature. (2) If the principal curvatures are given by a sum of functions of separated variables, there are no such hypersurfaces for $n{geq}4$, and for $n=3$, they are, up to M$ddot{o}$bius transformations, Dupin hypersurfaces with constant M$ddot{o}$bius curvature.