Polynomials with general C²-fibers are variables. I

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Suppose that X' is a smooth affine algebraic variety of dimension 3 with H₃(X')=0 which is a UFD and whose invertible functions are constants. Suppose that Z is 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. As a consequence we obtain the fact formulated in the title of the paper.