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 $K3$-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.
"Commuting families in skew fields and quantization of Beauville's fibration." Duke Math. J. 119 (2) 197 - 219, 15 August 2003. https://doi.org/10.1215/S0012-7094-03-11921-3