Complete Algebraic Vector Fields on Danielewski Surfaces

TL;DR

This paper classifies all complete algebraic vector fields on Danielewski surfaces, identifying their structure through preserved fibrations and providing explicit descriptions of these fibrations.

Contribution

It offers the first comprehensive classification of complete algebraic vector fields on Danielewski surfaces, including explicit descriptions of associated fibrations.

Findings

Classification of all complete algebraic vector fields on Danielewski surfaces

Explicit description of fibrations with fibers  and ^* on these surfaces

Analysis of regular functions with specific fiber types on Gizatullin surfaces

Abstract

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.