  In this paper we study the Brill Noether locus of rank 2, (semi)stable vector
bundles with at least two sections and of suitable degrees on a general
$\nu$-gonal curve. We classify its reduced components whose dimensions are at
least the corresponding Brill-Noether number. We moreover describe the general
member $\mathcal F$ of such components just in terms of extensions of line
bundles with suitable {\em minimality properties}, providing information on the
birational geometry of such components as well as on the very-ampleness of
$\mathcal F$.
