A characterization of domains in $\mathbf C^2$ with noncompact   automorphism group
  Let $D$ be a bounded domain in $\mathbf C^2$ with a non-compact group of holomorphic automorphisms. Model domains for $D$ are obtained under the hypothesis that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. 
Kaushal Verma