Integrability of connections with partially vanishing curvature

Let $$E \rightarrow B$$ be a vector bundle with a connection $$\nabla$$ and a local frame $$(e_1, \dots, e_n)$$. For any section $$V = V^a e_a$$ the connection can locally be written as $$\nabla V = \left(d V^a + \omega^a{}_b V^b\right) e_a \,,$$ where the connection 1-form is defined by $$\nabla e_a = e_b \omega^b{}_a$$. The corresponding curvature 2-form is given by $$\Omega = d \omega + \tfrac12 [\omega, \omega] \,.$$ It is well known that if $$\Omega = 0$$, it is possible to perform a gauge transformation (i.e. a change of basis) to set $$\omega = 0$$. (Of course this statement holds only locally, globally there might be additional topological obstructions.)

Now, let $$V$$ be a (local) section of $$E$$ such that $$\Omega^a{}_b V^b = 0 \,.$$ Is it possible to make a similar statement as before, i.e. to obtain $$\omega^a{}_b V^b = 0 \,,$$ by a suitable gauge transformation? I assume that this is in general not the case.

A sufficient condition for the previous statement seems to be the stronger condition $$\nabla V = 0 \,.$$ Is it possible to give a (weaker) necessary condition?