On complete generators of certain Lie algebras on Danielewski surfaces

We study the Lie algebra of polynomial vector fields on a smooth Danielewski surface of the form $xy=p(z)$ with $x,y,z\in \mathbb{C}$. We provide explicitly given generators to show that: 1. The Lie algebra of polynomial vector fields is generated by $6$ complete vector fields. 2. The Lie algebra of volume-preserving polynomial vector fields is generated by finitely many vector fields, whose number depends on the degree of the defining polynomial. 3. There exists a Lie sub-algebra generated by $4$ LNDs whose flows generate a group that acts infinitely transitively on the Danielewski surface. The latter result is also generalized to higher dimensions where $z\in {\mathbb{C}}^N$.