  Let $F\subseteq\mathbb{P}^3$ be a smooth quartic surface and let
$\mathcal{O}_F(h):=\mathcal{O}_{\mathbb{P}^3}(1)\otimes\mathcal{O}_F$. In the
present paper we classify locally free sheaves $\mathcal{E}$ of rank $2$ on $F$
such that $c_1(\mathcal{E})=\mathcal{O}_F(2h)$, $c_2(\mathcal{E})=8$ and
$h^1\big(F,\mathcal{E}(th)\big)=0$ for $t\in\mathbb{Z}$. We also deal with
their stability.
