Quantizations of the Hitchin and Beauville–Mukai integrable systems

Spectral transformation is known to set up a birational morphism between the Hitchin and Beauville–Mukai integrable systems. The corresponding phase spaces are: (a) the cotangent bundle of the moduli space of bundles over a curve $C$, and (b) a symmetric power of the cotangent surface $T^*(C)$. We conjecture that this morphism can be quantized, and we check this conjecture in the case where $C$ is a rational curve with marked points and rank 2 bundles. We discuss the relation of the resulting isomorphism of quantized algebras with Sklyanin's separation of variables.