Enriques surfaces with trivial Brauer map and involutions on hyperkähler 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 $\bar{X}$. We relate the fixed locus of this involution to certain Picard schemes of the noncommutative pair $(X,\mathcal{A})$, where $\mathcal{A}$ is an Azumaya algebra on $X$ defined by the nontrivial element in the Brauer group of $X$.