Standard isotrivial fibrations with pg=q=1

A smooth, projective surface S of general type is said to be a standard isotrivial fibration if there exists a finite group G acting faithfully on two smooth projective curves C and F so that S is isomorphic to the minimal desingularization of T:=(C×F)/G. If T is smooth then S=T is called a quasi-bundle. In this paper we classify the standard isotrivial fibrations with pg=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 pg=q=1 and .