Surfaces of general type with $p_g=q=1, \; K^2=8$ and bicanonical map of degree $2$

We classify the minimal algebraic surfaces of general type with $p_g=q=1, \; K^2=8$ and bicanonical map of degree $2$. It will turn out that they are isogenous to a product of curves, i.e. if $S$ is such a surface, then there exist two smooth curves $C, \; F$ and a finite group $G$ acting freely on $C \times F$ such that $S = (C \times F)/G$. We describe the $C, \; F$ and $G$that occur. In particular the curve $C$ is a hyperelliptic-bielliptic curve of genus $3$, and the bicanonical map $\phi$ of $S$ is composed with the involution $\sigma$ induced on $S$ by $\tau \times id: C \times F \longrightarrow C \times F$, where $\tau$ is the hyperelliptic involution of $C$. In this way we obtain three families of surfaces with $p_g=q=1, \; K^2=8$which yield the first-known examples of surfaces with these invariants. We compute their dimension and we show that they are three generically smooth, irreducible components of the moduli space $\mathcal{M}$ of surfaces with $p_g=q=1, \; K^2=8$. Moreover, we give an alternative description of these surfaces as double covers of the plane, recovering a construction proposed by Du Val.