  We give the classification of all complete algebraic vector fields on
Danielewski surfaces (smooth surfaces given by $xy=p(z)$). We use the fact that
for each such vector field there exists a certain fibration that is preserved
under its flow. In order to get the explicit list of vector fields a
classification of regular function with general fiber $\mathbb{C}$ or
$\mathbb{C}^*$ is required. In this text we present results about such
fibrations on Gizatullin surfaces and we give a precise description of these
fibrations for Danielewski surfaces.
