  Beilinson--Bernstein localisation relates representations of a Lie algebra
$\mathfrak{g}$ to certain $\mathcal{D}$-modules on the flag variety of
$\mathfrak{g}$. In [arXiv:2002.01540], examples of
$\mathfrak{sl}_2$-representations which correspond to $\mathcal{D}$-modules on
$\mathbb{CP}^1$ were computed. In this expository article, we give a
topological description of these and extended examples via the Riemann-Hilbert
correspondence. We generalise this to a full characterisation of
$\mathfrak{sl}_2$-representations which correspond to holonomic
$\mathcal{D}$-modules on $\mathbb{CP}^1$ with at most 2 regular singularities.
We construct further examples with more singularities and develop a computer
program for the computation of this correspondence in more general cases.
