We have that every holomorphic $\mathbb{CP}^1$-bundle on a compact Riemann surface admits a holomorphic section, as found in Compact Compact Surfaces of Barth, Peters and Van de Ven, for example.

If I add to this bundle a meromorphic flat connection (equivalently a foliation $\mathcal{F}$ which is transverse to the fibers except in finitely many fibers) is it possible to find a transverse section through the foliation $\mathcal{F}$?