  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 included in R and
such that R has transcendence degree 2 over K, then R is K-isomorphic to
K[X,Y,Z]/(XY-P(Z)) for some nonconstant polynomial P(Z) in K[Z].
