A complex manifold X X of dimension n n together with an ample vector bundle E E on it will be called a generalized polarized variety. The adjoint bundle of the pair ( X , E ) (X,E) is the line bundle K X + d e t ( E ) K_X + det(E) . We study the positivity (the nefness or ampleness) of the adjoint bundle in the case r := r a n k ( E ) = ( n − 2 ) r := rank (E) = (n-2) . If r ≥ ( n − 1 ) r\geq (n-1) this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski. If K X + d e t E K_X+detE is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map π : X ⟶ W \pi :X \longrightarrow W from X X onto a normal projective variety W W with connected fiber and such that K X + d e t ( E ) = π ∗ H K_X + det(E) = \pi ^*H , for some ample line bundle H H on W W . We describe those contractions for which d i m F ≤ ( r − 1 ) dimF \leq (r-1) . We extend this result to the case in which X X has log terminal singularities. In particular this gives Mukai’s conjecture 1 for singular varieties. We consider also the case in which d i m F = r dimF = r for every fiber and π \pi is birational.