Standard isotrivial fibrations with pg=q=1pg=q=1, II

A smooth, projective surface SS is called a standard isotrivial fibration   if there exists a finite group GG which acts faithfully on two smooth projective curves CC and FF so that SS is isomorphic to the minimal desingularization of T≔(C×F)/GT≔(C×F)/G. Standard isotrivial fibrations of general type with pg=q=1pg=q=1 have been classified in [F. Polizzi, Standard isotrivial fibrations with pg=q=1pg=q=1, J. Algebra 321 (2009),1600–1631] under the assumption that TT has only Rational Double Points as singularities. In the present paper we extend this result, classifying all cases where SS is a minimal model. As a by-product, we provide the first examples of minimal surfaces of general type with pg=q=1pg=q=1, KS2=5 and Albanese fibration of genus 3. Finally, we show with explicit examples that the case where SS is not minimal actually occurs.