On locally nilpotent derivations of k[X1,X2,Y]/( (Y)−X1X2)

We investigate the locally nilpotent derivations of the k-algebra B=k[X1,X2,Y]/( −X1X2),where k is a field of characteristic zero, X1,X2,Y are indeterminates over k and k[Y] is a nonconstant polynomial. We prove that a certain subgroup of Autk(B) acts transitively on the set of kernels of nonzero locally nilpotent derivations of B.