  The $\mathbb{Z}$-genus of a link $L$ in $S^3$ is the minimal genus of a
locally flat, embedded, connected surface in $D^4$ whose boundary is $L$ and
with the fundamental group of the complement infinite cyclic. We characterise
the $\mathbb{Z}$-genus of boundary links in terms of their single variable
Blanchfield forms, and we present some applications. In particular, we show
that a variant of the shake genus of a knot, the $\mathbb{Z}$-shake genus,
equals the $\mathbb{Z}$-genus of the knot.
