On $U(\mathfrak{h})$-free modules over $\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }n)$

We study two categories of $U(\mathfrak{h})$-free $\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }n)$-modules of total rank 2: ${\mathcal{M}}_{\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }n)}(2)$, whose objects are free of rank 2 over $U(\mathfrak{h})$ which are not necessarily ${\mathbb{Z}}_2$-graded, and ${\mathcal{M}}_{\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }n)}(1\mathrm{\mid }1)$, whose objects are supermodules with even and odd parts each isomorphic to $U(\mathfrak{h})$. For $\mathfrak{s}\mathfrak{l}(m\mathrm{\mid }1)$ we give a complete classification in both categories, and we prove that for $m,n\ge 2$ both categories are empty.