Affine surfaces with trivial Makar-Limanov invariant

We study the class of 2-dimensional affine k-domains R satisfying ML(R)=k, where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many variables over a field of characteristic zero. If ML(R)=K for some field K R such that trdegKR=2, then R is K-isomorphic to K[X,Y,Z]/(XY−P(Z)) for some nonconstant P(Z) K[Z].