Moduli spaces of vector bundles on a curve and opers

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\ge 2$, and let $\xi$ be a holomorphic line bundle on $X$ with ${\xi }^{\otimes 2}={\mathcal{O}}_X$. Fix a theta characteristic $\mathbb{L}$ on $X$. Let ${\mathcal{M}}_X(r,\xi )$ be the moduli space of stable vector bundles $E$ on $X$ of rank $r$ such that ${\bigwedge }^{r}E=\xi$ and $H^0(X,E\otimes \mathbb{L})=0$. Consider the quotient of ${\mathcal{M}}_X(r,\xi )$ by the involution given by $E⟼E^{\ast }$. We construct an algebraic morphism from this quotient to the moduli space of $\mathrm{S}\mathrm{L}(r,\mathbb{C})$ opers on $X$. Since $\dim {\mathcal{M}}_X(r,\xi )$ coincides with the dimension of the moduli space of $\mathrm{S}\mathrm{L}(r,\mathbb{C})$ opers, it is natural to ask about the injectivity and surjectivity of this map.