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| The Sitnikov problem is a restricted three body problem where the primaries bodies are moving in a symmetric way on elliptic orbits and such that the infinitesimal body is moving in the orthogonal direction of them and passing by the center of mass $O$. The position $z(t)$ of the infinitesimal body relative to $O$ satisfies the following differential equation in appropriate units,
\begin{equation}\label{sp}
\ddot{z}+\frac{z}{(z^{2}+r(t,e)^{2})^{3/2}}=0,
\end{equation}
where $e\in [0,1[$ is the eccentricity of the primaries and r(t,e) measures the distance between any them and $O.$
This problem became important in $1960$ when Stnikov used it to show, for the first time, the possibility of the existence of oscillatory motion in the $3$-body problem. Many contributions have been made since then like for instance a paper series of Alekseev in $1968$-$1969$, where he proved that all of the possible combinations of final motions in the sense of Chazy were realized in the Sitnikov problem. To the last decade of the past century and at the present one the families of symmetric periodic orbits of the Sitnikov problem (depending of the eccentricity) have been studied by several authors like Belbruno and Llibre \cite{Belbruno}, Jimenez-Lara and Escalona Buen-Dia \cite{Jimenez}, Ortega and Llibre \cite{Llibre-Ortega}, Ortega and Rivera \cite{Ortega-Rivera}, and many others. Worth mentioning the results of Ortega-Llibre where the authors have obtained analytically and by topological methods the continuation of some periodic orbits for $e=0$ to all values of $e\in [0,1[$. However it seems that nothing has been said about the stability properties of some part of these branches. In this talk we present some strategies for quantifying the periodic even elliptic segment of the branch emerging from the center of mass of the circular problem ($e=0$). These techniques can be also exploited to estimate the size of a stable segment of symmetric and periodic solutions in oscillators like forced pendulum, obtaining similar results in the line of \cite{zhang}. For this purpose the results in \cite{nunez} could be useful.
Keywords: $3$-body problem, Sitnikov problem,
periodic orbits, global continuation, ellipticity.
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