  We introduce a family of rank-one local systems in the category of twisted
$\mathcal{D}$-modules on a certain subvariety isomorphic to
${\mathbb{G}_{\text{m}}}^2$ of the affine flag variety of $\text{SL}_2$. We
then give a criterion for these local systems, in terms of their parameters, to
extend cleanly in the sense of $\mathcal{D}$-modules.
