On projective threefolds with two-dimensional space of vanishing cycles

We obtain a complete list of smooth projective threefolds over $\mathbb{C}$ for which the dimension of the space of vanishing cycles (in $H^2$ of the smooth hyperplane section) equals $2$. We also obtain a complete list of rank 2 very ample vector bundles $E$ on smooth projective surfaces with $c_2(E)=3$.