  We study smooth quadric surfaces in the Pfaffian hypersurface in
$\mathbb{P}^{14}$ parameterising $6 \times 6$ skew-symmetric matrices of rank
at most 4, not intersecting the Grassmannian $\mathbb{G}(1,5)$. Such surfaces
correspond to quadratic systems of skew-symmetric matrices of size 6 and
constant rank 4, and give rise to a globally generated vector bundle $E$ on the
quadric. We analyse these bundles and their geometry, relating them to linear
congruences of lines in $\mathbb{P}^5$.
