  A smooth algebraic surface $S$ is said to be \emph{isogenous to a product of
unmixed type} if there exist two smooth curves $C, F$ and a finite group $G$,
acting faithfully on both $C$ and $F$ and freely on their product, so that
$S=(C \times F)/G$. In this paper we classify the surfaces of general type with
$p_g=q=1$ which are isogenous to an unmixed product, assuming that the group
$G$ is abelian. It turns out that they belong to four families, that we call
surfaces of type $I, II, III, IV$. The moduli spaces $\mathfrak{M}_{I},
  \mathfrak{M}_{II}, \mathfrak{M}_{IV}$ are irreducible, whereas
$\mathfrak{M}_{III}$ is the disjoint union of two irreducible components. In
the last section we start the analysis of the case where $G$ is not abelian, by
constructing several examples.
