Complete algebraic vector fields on affine surfaces

Let $\text{AAutH}(X)$ be the subgroup of the group $\text{AutH}(X)$ of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces $X$ quasi-homogeneous under $\text{AAutH}(X)$ in terms of the dual graphs of the boundaries $\bX \setminus X$oftheirSNC-completions$\bX$.