Let $X$ be a projective curve and $G$ be a semisimple Lie group. There is a theorem roughly stating that there exists an isomorphism between the moduli space of principal $G$-bundles on $X$ and the moduli space of $G$-affine opers on $X$. Moreover, this isomorphism is compatible with the action of Heisenberg subgroup of the loop group $LG$ on the former and the action of generalized Drinfeld--Sokolov hierarchy on the latter.
For $X$ a hyperelliptic curve and $G=SL_2$, this theorem is essentially equivalent to Krichever's classical construction of algebro-geometric solutions of KdV hierarchy (and the compatibility condition expresses the fact that KdV flow linearizes on the Jacobian variety of $X$).
My question is: is there anything known about analogue of this theorem for $q$-difference operators (as opposed to opers, which are basically differential operators)? If such an analogue exists, it would presumably entail a $q$-version of Drinfeld--Sokolov hierachy.