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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