  This paper classifies surfaces of general type $S$ with $p_g=q=1$ having an
involution $i$ such that $S/i$ has non-negative Kodaira dimension and that the
bicanonical map of $S$ factors through the double cover induced by $i.$
  It is shown that $S/i$ is regular and either: a) the Albanese fibration of
$S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ 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.
