Aggressive geometry – Can a holomorphic vector beam on a compact Riemann surface be defined by a single transition function?

It is known that any holomorphic beam of any rank on a non-compact Riemann surface is trivial. Evidence of this is given in Forster's "Conferences on Riemann Surfaces", section 30.

Let $ E $ to be a holomorphic vector beam on a compact surface of Riemann $ X $ with gauge group $ G $. A consequence of the above theorem is the restriction $ E | _ {X – {p }} $ for any point $ p in X $ is a trivial package. So $ E $ can be recovered by specifying the transition function $ g: D cap (X – {p }) rightarrow $ G or $ D $ is a small disk containing $ p $.

Is it correct? If no, could you give a counterexample? I am mainly interested in learning the space of holomorphic beam modules $ X $ concretely, for example by using transition functions.