Complete Algebraic Vector Fields on Danielewski Surfaces

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 ${\mathbb{C}}^{\ast }$ 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.