  Let k be an algebraically closed field of characteristic zero, D a locally
nilpotent derivation on the polynomial ring k[X_1, X_2,X_3,X_4] and A the
kernel of D. A question of M. Miyanishi asks whether projective modules over A
are necessarily free. Implicit is a subquestion: whether the Grothendieck group
K_0(A) is trivial.
  In this paper we shall demonstrate an explicit k[X_1]-linear fixed point free
locally nilpotent derivation D of k[X_1,X_2, X_3, X_4] whose kernel A has an
isolated singularity and whose Grothendieck group K_0(A) is not finitely
generated; in particular, there exists an infinite family of pairwise
non-isomorphic projective modules over the kernel A.
  We shall also show that, although Miyanishi's original question does not have
an affirmative answer in general, suitably modified versions of the question do
have affirmative answers when D annihilates a variable. For instance, we shall
establish that in this case the groups G_0(A) and G_1(A) are indeed trivial.
Further, we shall see that if the above kernel A is a regular ring, then A is
actually a polynomial ring over k; in particular, by the Quillen-Suslin
theorem, Miyanishi's question has an affirmative answer.
  Our construction involves rings defined by the relation u^mv=F(z,t), where
F(Z,T) is an irreducible polynomial in k[Z,T]. We shall show that a necessary
and sufficient condition for such a ring to be the kernel of a k[X_1]-linear
locally nilpotent derivation D of a polynomial ring k[X_1,...,X_4] is that F
defines a polynomial curve.
