Commuting families in skew fields and quantization of Beauvilles fibration

We construct commuting families in fraction fields of symmetric powers of algebras. The classical limit of this construction gives Poisson commuting families associated with linear systems. In the case of a K-surface S, they correspond to Lagrangian fibrations introduced by A. Beauville. When S is the canonical cone of an algebraic curve C, we construct commuting families of differential operators on symmetric powers of C, quantizing the Beauville systems.