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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 theessential range of that operator, there exists a function $u$ of specialbounded variation satisfying  \[  \mathcal A u(x)= f(x) \qquad \text{almost everywhere}.  \] This extends a result of G. Alberti for gradients on $\mathbf R^N$. Inparticular, for $0 \le m < N$, it is shown that every integrable $m$-vectorfield is the absolutely continuous part of the boundary of a normal$(m+1)$-current.

A Lebesgue-Lusin property for linear operators of first and second order