Locally nilpotent derivations on affine surfaces with a ${\text{C}}^{\ast }$-action

We give a classification of normal affine surfaces admitting an algebraic group action with an open orbit. In particular an explicit algebraic description of the affine coordinate rings and the defining equations of such varieties is given. By our methods we recover many known results, e.g. the classification of normal affine surfaces with a `big’ open orbit of Gizatullin and Popov or some of the classification results of Danilov-Gizatullin, Bertin and others.