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Let $V$ be an irreducible complex analytic space of dimension two with normalsingularities and $\vr:\mathbb{C^*}\times V\to V$ a holomorphic action of thegroup $\mathbb{C^*}$ on $V$. Denote by $\fa_\vr$ the foliation on $V$ inducedby $\vr$. The leaves of this foliation are the one-dimensional orbits of $\vr$.%and its singularities are the fixed points of $\vr$. We will assume that thereexists a \emph{dicritical} singularity $p\in V$ for the $\bc^*$-action, i.e.for some neighborhood $p\in W\subset V$ there are infinitely many leaves of$\mathcal {F}_\vr|_{W}$ accumulating only at $p$. The closure of such a localleaf is an invariant local analytic curve called a \emph{separatrix} of$\mathcal{F}_\vr$ through $p$. In \cite{Orlik} Orlik and Wagreich studied the2-dimensional affine algebraic varieties embedded in $\mathbb{C}^{n+1}$, withan isolated singularity at the origin, that are invariant by an effectiveaction of the form $\sigma_Q(t,(z_{0},...,z_{n}))=(t^{q_{0}}z_{0},...,t^{q_{n}}z_{n})$ where $Q=(q_0,...,q_n) \in\mathbb N^{n+1}$, i.e. all $q_{i}$are positive integers. Such actions are called \emph{good} actions. Inparticular they classified the algebraic surfaces embedded in $\mathbb{C}^{3}$endowed with such an action. It is easy to see that any good action on asurface embedded in $\mathbb{C}^{n+1}$ has a dicritical singularity at$0\in\mathbb{C}^{n+1}$. Conversely, it is the purpose of this paper to showthat good actions are the models for analytic $\mathbb{C^*}$-actions on Steinanalytic spaces of dimension two with a dicritical singularity.

C^*- Actions on Stein analytic spaces with isolated singularities