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May 29, 2020 at 18:41 comment added Andy Sanders Indeed. I'm glad I could say something useful.
May 29, 2020 at 18:39 comment added mtraube @AndySanders, nice, thanks for the references! The beginning of section 2 in the paper by Beilinson-Drinfeld directly answers the first question with the additional assumption that the differential operator has nowhere vanishing principal symbol. So representing the vector bundle locally on a coordinate open as $\mathcal{D}_1(U)/ \mathcal{D}_1P(U)$ the differential operators $P(U)$ only glue to a differential operator with non vanishing principal symbol if the bundle has in fact an oper structure.
May 29, 2020 at 17:41 comment added Andy Sanders See also section 2 of the notes arxiv.org/pdf/math/0501398.pdf of Beilinson-Drinfeld where they also basically answer your question.
May 29, 2020 at 17:39 comment added Andy Sanders I think the answer to your question should be contained, or at least intimated, in these notes of Richard Wentworth: arxiv.org/abs/1402.4203.
May 29, 2020 at 17:07 history asked mtraube CC BY-SA 4.0