Algebraic torus actions on affine algebraic surfaces

We investigate smooth affine surfaces with Gm-actions and A∗1-fibrations. We consider a smooth affine surface with an untwisted A∗1-fibration over the affine line, and obtain a characterization of the fibration is unique in terms of the boundary divisor of the surface. As an application of the characterization, we know that a smooth affine surface admitting two independent Gm-actions is isomorphic to A∗1×A∗1 or A1×A∗1 unless S has an A1-fibration over A1.