Enriques surfaces with trivial Brauer map and involutions on hyperk\"ahler manifolds

Let X be an Enriques surface. Using Beauville's result about the triviality of the Brauer map of X, we define a new involution on the category of coherent sheaves on the canonically covering K3 surface X̄. We relate the fixed locus of this involution to certain Picard schemes of the noncommutative pair (X,A), where A is an Azumaya algebra on X defined by the nontrivial element in the Brauer group of X.