On surfaces with $p_g=q=1$ and non-ruled bicanonical involution

This paper classifies surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S\mathrm{/}i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i\mathrm{.}$ It is shown that $S\mathrm{/}i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S\mathrm{/}i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is also constructed.