Derivations of R[X,Y,Z] with a slice

The question under consideration is whether every locally nilpotent R-derivation of R[X,Y,Z] with a slice has kernel A generated by two elements over R, where R is a polynomial ring over a field of characteristic zero. Theorem 1.1 gives a fundamental property of such kernels, namely, that A is an A2-fibration over R. While it is an open question whether every A2-fibration over R is trivial, the property of A asserted in the theorem is necessary to the condition that A is a polynomial ring in two variables over R. The last section of the paper presents a family of examples θn (n⩾1) which are quite simple to define, but whose status relative to the kernel question is not known.