  We study certain pencils of del Pezzo surfaces generated by a smooth del
Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a
weighted projective space P and an appropriate multiple of a hyperplane H. Our
main observation is that every minimal model program relative to the morphism
lifting such pencil on a suitable resolution of its indeterminacies preserves
the open subset P H \^a A^3. As an application, we obtain projective
completions of A^3 into del Pezzo fibrations over P^1 of every degree less or
equal to 4. We also obtain completions of A^3 into Mori conic bundles, whose
restrictions to A^3 are twisted C*-fibrations over A^2 .
