Complete intersection surfaces with trivial Makar-Limanov invariant

We develop a framework for studying normal rational surfaces which are connected at infinity and admit an A1-fibration. As an application, we obtain the following result. Let S be an affine surface over a field of characteristic zero. If S is a complete intersection and has trivial Makar-Limanov invariant, then S is isomorphic to a hypersurface of affine 3-space with equation XZ=P(Y), for some nonconstant polynomial P(Y) in one variable.