  The rectilinear elliptic restricted Three Body Problem (TBP) is the limiting
case of the elliptic restricted TBP when the motion of the primaries is
described by a Keplerian ellipse with eccentricity $e'=1$, but the collision of
the primaries is assumed to be a non-singular point. The rectilinear model has
been proposed as a starting model for studying the dynamics of motion around
highly eccentric binary systems. Broucke (1969) explored the rectilinear
problem and obtained isolated periodic orbits for mass parameter $\mu=0.5$
(equal masses of the primaries). We found that all orbits obtained by Broucke
are linearly unstable. We extend Broucke's computations by using a finer search
for symmetric periodic orbits and computing their linear stability. We found a
large number of periodic orbits, but only eight of them were found to be
linearly stable and are associated with particular mean motion resonances.
These stable orbits are used as generating orbits for continuation with respect
to $\mu$ and $e'<1$. Also, continuation of periodic solutions with respect to
the mass of the small body can be applied by using the general TBP. FLI maps of
dynamical stability show that stable periodic orbits are surrounded in phase
space with regions of regular orbits indicating that systems of very highly
eccentric orbits can be found in stable resonant configurations. As an
application we present a stability study for the planetary system HD7449.
