Danielewski–Fieseler surfaces

We study a class of normal affine surfaces, with additive group actions, which contains in particular the Danielewski surfaces in A³ given by the equations xⁿz = P(y), where P is a nonconstant polynomial with simple roots. We call them Danielewski--Fieseler surfaces. We reinterpret a construction of Fieseler to show that these surfaces appear as the total spaces of certain torsors under a line bundle over a curve with an r fold point. We classify Danielewski-Fieseler surfaces through labelled rooted trees attached to such a surface in a canonical way. Finally, we characterize those surfaces which have a trivial Makar-Limanov invariant in terms of the associated trees.