Opers on transversely holomorphic foliations

For real codimension two smooth foliations, we study the transversely complex structures, transversely complex projective structures and transverse opers associated to the foliation. We prove a uniqueness theorem for the transversely holomorphic $\mathbb{C}{\mathbb{P}}^1$–bundle naturally associated with a transversely complex projective structures. A similar uniqueness theorem is proved for the transversely holomorphic filtered $\mathbb{C}{\mathbb{P}}^{r-1}$–bundle naturally associated to a transverse $\mathrm{P}\mathrm{G}\mathrm{L}(r,\mathbb{C})$–oper.