  Let $X$ be a Fano threefold with index $i_X$ and fundamental line bundle
$\mathcal O_X(h)$. We classify $\mu$-semistable rank two bundles $\mathcal E$
on $X$ with $c_1(\mathcal E)=0$, $h^0(\mathcal E) \ne 0$ and $h^1(\mathcal
E(-\lceil\frac{i_X}{2}\rceil h))=0$.
