  The affine line minus one point is the underlying space of the algebraic
torus of dimension one. However the fibration of an affine algebraic threefold
by the affine line minus one point is not always the quotient morphism of the
threefold by the algebraic torus. We investigate the structure of affine
algebraic threefolds having fibrations by the affine line minus one point. Our
results cover not only algebro-geometric but also topological properties of the
threefolds. We also consider when the fibration by the affine line minus one
point becomes the quotient morphism by an algebraic torus of dimension one.
