Free 2-step nilpotent Lie algebras and indecomposable modules

Given an algebraically closed field $F$ of characteristic 0 and an $F$-vector space $V$, let $L(V)=V\oplus\Lambda^2(V)$ denote the free 2-step nilpotent Lie algebra associated to $V$. In this paper, we classify all uniserial representations of the solvable Lie algebra $\mathfrak g=\langle x\rangle\ltimes L(V)$, where $x$ acts on $V$ via an arbitrary invertible Jordan block.