On polynomials in three variables annihilated by two locally nilpotent derivations

Let B be a polynomial ring in three variables over an algebraically closed field k of characteristic zero. We are interested in irreducible polynomials f∈B satisfying the following condition: there exist nonzero locally nilpotent derivations such that ker(D1)≠ker(D2) and D1(f)=0=D2(f). The main result asserts that a nonconstant polynomial f∈B satisfies the above requirement if and only if its “generic fiber” k(f)⊗k[f]B is isomorphic, as an algebra over the field K=k(f), to K[X,Y,Z]/(XY−φ(Z)) for some nonconstant φ(Z)∈K[Z].