computing the Clifford index of $C$, that is, satisfying $\mbox{deg}(A)-2h^0(C, A)+2=\mathrm{Cliff}(C)$, where $h^0(C, A)\geq 2$. We set $E:=A\oplus A$ and note that $\gamma(E)=\gamma(A)=\mathrm{Cliff}(C)$.

An attempt to determine $\gamma_2(C)$ for a general curve $[C]\in \cM_g$, can be linked to an older conjecture of Mercat [@Me]. As already mentioned in the Introduction, for a bundle $E\in \widetilde{\cU}_C(2, d)$ with $\mathrm{Cliff}(C) + 2 \leq \mu(E) \leq 2g-4 - \mathrm{Cliff}(C)$, it was predicted that $$h^0(C, E) \leq 2+\mu(E) - \mathrm{Cliff}(C).$$ As pointed out in [@LN], a consequence of (\[mercat1\]) and (\[mercat2\]) is the equality $\gamma_2(C)=\mathrm{Cliff}(C)$. A positive answer to Mercat’s question, would show that, from the point of view of Clifford theory, special rank $2$ vector bundles are determined by special classical linear series. Inequalities (\[mercat1\]), (\[mercat2\]) hold trivially when $h^0(C, E) \leq 3$, thus one may assume that $h^0(C, E) \geq 4$. The following observation is essentially contained in [@Me]. We choose to make it explicit in order to make the bounds in (\[mercat1\]) and (\[mercat2\]) transparent to ourselves:

Let $E\in \cU_C(2, d)$ with $\mu(E)\leq g-1$. If $E$ contains a sub-pencil, then (\[mercat1\]) holds.

Suppose that the vector bundle $E$ fits into an exact sequence $$0 {\rightarrow}A {\rightarrow}E {\rightarrow}A' {\rightarrow}0,$$ with $A$ a 