Adjoint bundles of ample and spanned vector bundles on algebraic surfaces

Let E be an ample and spanned vector bundle of rank r \geq 2 on a smooth complex propjective surface X and set L = \det E. We show that K_X+L is spanned except for (X,E)=(P^2, O(1)^{\oplus 2}) and we describe all pairs (X,E) for which K_X+det E fais to be very ample.