The classification of isotrivial fibred surfaces with \(p_g=q=2\)

An isotrivial fibred surface is a smooth projective surface endowed with a morphism onto a curve such that all the smooth fibres are isomorphic to each other. The first goal of this paper is to classify the isotrivial fibred surfaces with \(p_g=q=2\) completing and extending a result of Zucconi. As an important byproduct, we provide new examples of minimal surfaces of general type with \(p_g=q=2\) and \(K^2=4,5\) and a first example with \(K^2=6\).