Retracts That Are Kernels of Locally Nilpotent Derivations

Let k be a field of characteristic zero and B a k-domain. Let R be a retract of B being the kernel of a locally nilpotent derivation of B. We show that if B = R circle plus I for some principal ideal I (in particular, if B is a UFD), then B = R-[1], i.e., B is a polynomial algebra over R in one variable. It is natural to ask that, if a retract R of a k-UFD B is the kernel of two commuting locally nilpotent derivations of B, then does it follow that B approximately equal to R-[2]? We give a negative answer to this question. The interest in retracts comes from the fact that they are closely related to Zariski's cancellation problem and the Jacobian conjecture in affine algebraic geometry.