  Since their introduction by Beilinson-Drinfeld \cite{BD,Opers1}, opers have
seen several generalizations. In \cite{BSY} a higher rank analog was studied,
named {generalized $B$-opers}, where the successive quotients of the oper
filtration are allowed to have higher rank and the underlying holomorphic
vector bundle is endowed with a bilinear form which is compatible with both the
filtration and the oper connection. Since the definition didn't encompass the
even orthogonal groups, we dedicate this paper to study generalized $B$-opers
whose structure group is ${\rm SO}(2n,\mathbb{C})$, and show their close
relationship with geometric structures on a Riemann surface.
