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?