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We consider fibrations by affine lines on smooth affine surfaces obtained ascomplements of smooth rational curves $B$ in smooth projective surfaces $X$defined over an algebraically closed field of characteristic zero. We observethat except for two exceptions, these surfaces $X \setminus B$ admit infinitelymany families of $\mathbb{A}^1$-fibrations over the projective line withirreducible fibers and a unique singular fiber of arbitrarily largemultiplicity. For $\mathbb{A}^1$-fibrations over the affine line, we give a newand essentially self-contained proof that the set of equivalence classes ofsuch fibrations up to composition by automorphisms at the source and target isfinite if and only if the self-intersection number of $B$ in $X$ is less thanor equal to 6.

Fibrations by affine lines on rational affine surfaces with irreducible boundaries