  This paper concerns (Lorentz-)Darboux transformations in the
Lorentz-Minkowski plane $\mathbb{R}^{1,1}$. We use the Penrose diagram for
conformal compactification and show some unique properties of Darboux
transformations of spacelike curves in $\mathbb{R}^{1,1}$, especially with
regard to singularities and blowup. Finally, we make some comments on Darboux
transformation of type-changing curves.
