We know that we have an alternative way to define a complex structure on manifold, by means of an integrable almost complex structure. The two points of view are equivalent, this is the content of the Newlander-Nirenberg theorem (which is very difficult to prove).
I think that there is the same kind of notion for vector bundles (just replace the bundle map between the tangent bundle and itself by a bundle map from your vector bundle and itself). My question is:
Do we have also an equivalence theorem in the spirit of Newlander-Nirenberg for general vector bundles?
If somebody could give me some reference towards an article, it would be great.