Ample vector bundles with sections vanishing on surfaces of Kodaira dimension zero

Let E be an ample vector bundle of rank r \geq 2 on a compact complex manifold X of dimension n=r+2, having a section whose zero locus is a smooth surface Z. Triplets (X,E,Z) as above are investigated under the assumption that Z has Kodaira dimension zero. It turns out that either X is a P^{n-2}-bundle over a smooth surface, E restricts to every fiber as n-2 copies of O_P(1), and the P-bundle projection restricted to Z is a birational morphism contracting some exceptional curves, or, up to contracting some (-1)-hyperplanes, X is a Fano manifold with -K_X = det E, in which case Z is a K3 surface.