UFDs with commuting linearly independent locally nilpotent derivations

In this paper we study affine K-UFDs of transcendence degree n without nonconstant units, having n−1 commuting linearly independent locally nilpotent K-derivations. We prove in case n=2, and K algebraically closed of characteristic zero, that such rings are polynomial rings in two variables over K. We then show that the commuting derivations Conjecture is equivalent to a weak version of the Abhyankar-Sathaye Conjecture.