  We give a general structure theorem for affine A 1-fibrations on smooth
quasi-projective surfaces. As an application, we show that every smooth A
1-fibered affine surface non-isomorphic to the total space of a line bundle
over a smooth affine curve fails the Zariski Cancellation Problem. The present
note is an expanded version of a talk given at the Kinosaki Algebraic Geometry
Symposium in October 2019.
