  Let $\AAutH (X)$ be the subgroup of the group $\AutH (X)$ 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 $\AAutH (X)$ in terms of the dual graphs of the boundaries $\bX \setminus
X$ of their SNC-completions $\bX$.
