On certain classes of triangular derivations of <i>R</i> [ <i>X, Y, Z</i> ] with three-generated kernels

Let [Formula: see text] be a Noetherian UFD containing [Formula: see text] and [Formula: see text]. We show that, under certain conditions, the kernel of the following type of [Formula: see text]-derivations [Formula: see text] of [Formula: see text] is generated by at most [Formula: see text] elements over [Formula: see text]: a) [Formula: see text] is an extension of an [Formula: see text]-elementary derivation [Formula: see text] of [Formula: see text] such that [Formula: see text] or [Formula: see text] is a homogeneous polynomial in [Formula: see text] and [Formula: see text] with respect to the standard grading. b) [Formula: see text] is a nice [Formula: see text]-derivation satisfying [Formula: see text] and [Formula: see text] form a regular sequence or are comaximal in [Formula: see text]. c) [Formula: see text] is triangular monomial. d) [Formula: see text] is an extension of an irreducible triangular [Formula: see text]-derivation [Formula: see text] of [Formula: see text] such that [Formula: see text].