Fixed point free locally nilpotent derivations of A2-fibrations

Let R be a regular ring containing Q and let A be an A2-fibration over R. We prove in this paper that every fixed point free locally nilpotent R-derivation of A has a slice. As a consequence, an A2-fibration over a factorial regular ring R is trivial if and only if it has a fixed point free locally nilpotent R-derivation. Our results answer a question raised by Freudenburg (2009) [11, Question 2] which actually was the main motivation of this work.