  Let $V$ be an irreducible complex analytic space of dimension two with normal
singularities and $\vr:\mathbb{C^*}\times V\to V$ a holomorphic action of the
group $\mathbb{C^*}$ on $V$. Denote by $\fa_\vr$ the foliation on $V$ induced
by $\vr$. 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 there
exists 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 local
leaf is an invariant local analytic curve called a \emph{separatrix} of
$\mathcal{F}_\vr$ through $p$. In \cite{Orlik} Orlik and Wagreich studied the
2-dimensional affine algebraic varieties embedded in $\mathbb{C}^{n+1}$, with
an isolated singularity at the origin, that are invariant by an effective
action 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. In
particular 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 a
surface 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 show
that good actions are the models for analytic $\mathbb{C^*}$-actions on Stein
analytic spaces of dimension two with a dicritical singularity.
