Polynomials with general C²-fibers are variables

Let X′ be a complex affine algebraic threefold with H₃(X′) = 0 which is a UFD and whose invertible functions are constants. Let Z be a Zariski open subset of X′ which has a morphism p : Z → U into a curve U such that all fibers of p are isomorphic to C². We prove that X′ is isomorphic to C³ iff none of irreducible components of X′ \ Z has non-isolated singularities. Furthermore, if X′ is C³ then p extends to a polynomial on C³ which is linear in a suitable coordinate system. This implies the fact formulated in the title of the paper.