  Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$
the module of $\mathcal{A}$-derivations view as the set of polynomial vector
fields which possess $\mathcal{A}$ as an invariant set. We first characterize
polynomial vector fields having an infinite number of invariant lines. Then we
prove that the minimal degree of polynomial vector fields fixing only a finite
set of lines in $\mathcal{D}(\mathcal{A})$ is not determined by the
combinatorics of $\mathcal{A}$.
