Simple ${\mathfrak{s}\mathfrak{l}}_2$-modules that are torsion free $U(\mathfrak{h})$-modules of rank $1$

We provide an explicit classification of all simple ${\mathfrak{s}\mathfrak{l}}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2)$.