  This paper is aimed to investigate the strong $L^p$-closure
$L_{\mathbb{Z}}^p(B)$ of the vector fields on the open unit ball
$B\subset\mathbb{R}^3$ that are smooth up to finitely many integer point
singularities. First, such strong closure is characterized for arbitrary
$p\in[1,+\infty)$. Secondly, it is shown what happens if the integrability
order $p$ is large enough (namely, if $p\ge 3/2$). Eventually, a decomposition
theorem for elements in $L_{\mathbb{Z}}^1(B)$ is given, conveying information
about the possibility of connecting the singular set of such vector fields by a
mass-minimizing, integer 1-current on $B$ with finite mass.
