  A smooth, projective surface $S$ of general type is said to be a
\emph{standard isotrivial fibration} if there exist a finite group $G$ which
acts faithfully on two smooth projective curves $C$ and $F$ so that $S$ is
isomorphic to the minimal desingularization of $T:=(C \times F)/G$. If $T$ is
smooth then $S=T$ is called a $\emph{quasi-bundle}$. In this paper we classify
the standard isotrivial fibrations with $p_g=q=1$ which are not quasi-bundles,
assuming that all the singularities of $T$ are rational double points. As a
by-product, we provide several new examples of minimal surfaces of general type
with $p_g=q=1$ and $K_S^2=4,6$.
