Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures

We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $\mu$ satisfying $\mathcal{A}\mu=0$. When $\mathcal{A}$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of $\mu$. Finally, we show that operators in the above class satisfy a Lusin-type property.