  A smooth, projective surface $S$ is called a $\emph{standard isotrivial
fibration}$ if there exists 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 $T:=(C \times F)/G$. Standard isotrivial fibrations of
general type with $p_g=q=1$ have been classified in \cite{Pol07} under the
assumption that $T$ has only Rational Double Points as singularities. In the
present paper we extend this result, classifying all cases where $S$ is a
minimal model. As a by-product, we provide the first examples of minimal
surfaces of general type with $p_g=q=1$, $K_S^2=5$ and Albanese fibration of
genus 3. Finally, we show with explicit examples that the case where $S$ is not
minimal actually occurs.
