  We prove that for a homogeneous linear partial differential operator
$\mathcal A$ of order $k \le 2$ and an integrable map $f$ taking values in the
essential range of that operator, there exists a function $u$ of special
bounded variation satisfying
  \[
  \mathcal A u(x)= f(x) \qquad \text{almost everywhere}.
  \] This extends a result of G. Alberti for gradients on $\mathbf R^N$. In
particular, for $0 \le m < N$, it is shown that every integrable $m$-vector
field is the absolutely continuous part of the boundary of a normal
$(m+1)$-current.
