  The \textit{Enriched Riemann Sphere} $\C P_*^1$ is $\C P^1$ plus a set of
\textit{infinitesimals}, having the Newton-Puiseux field $\F$ as coordinates.
Complex Analysis is extended to the $\F$-\textit{Analysis}
(\textit{Newton-Puiseux Analysis}). The classical \textit{Morse Stability
Theorem} is also extended; the \textit{stability idea} is used to formulate an
\textit{equi-singular deformation theorem} in $\C\{x,y\}(=\mathcal{O}_2)$.
