  In this work we introduce a planar restricted four-body problem where a
massless particle moves under the gravitational influence due to three bodies
following the eight figure choreography, and we explore some symmetric periodic
orbits of this system which turns out to be non autonomous. We demonstrate that
certain families of involutions are reversing symmetries of certain N-body
problem. We consider a particular case of such families of reversing symmetries
and we use it to study both theoretically and numerically certain type of
symmetric periodic orbits of this system. The symmetric periodic orbits
(initial conditions) were determined by means of solving some boundary value
problems.
