  Let $F$ be an algebraically closed field and consider the Lie algebra
${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$, where $\mathrm{ad}\, x$
acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Refer to a
${\mathfrak g}$-module as admissible if $[{\mathfrak g},{\mathfrak g}]$ acts
via nilpotent operators on it, which is automatic if $\mathrm{char}(F)=0$. In
this paper we classify all indecomposable ${\mathfrak g}$-modules $U$ which are
admissible as well as uniserial, in the sense that $U$ has a unique composition
series.
