Families of symmetric periodic orbits in the three body problem and the figure eight F. J. Mu˜ noz-Almaraz † ,J.Gal´an ‡ and E. Freire ‡ † Departamento de Matem´ atica Aplicada y Computaci´on. Facultad de Ciencias. Universidad de Valladolid. 47005 Valladolid, Spain. ‡ Departamento de Matem´ atica Aplicada II. Escuela Superior de Ingenieros. Universidad de Sevilla. 41092 Sevilla, Spain. Monograf´ ıas de la Real Academia de Ciencias de Zaragoza. 25: 229–240, (2004). Abstract In this paper we show a technique for the continuation of symmetric periodic or- bits in systems with time-reversal symmetries. The geometric idea of this technique allows us to generalize the “cylinder” theorem for this kind of systems. We state the main theoretical result without proof (to be published elsewhere). We focus on the application of this scheme to the three body problem (TBP), taking as starting point the figure eight orbit [3] to find families of symmetric periodic orbits. Key words and expressions: Hamiltonian and conservative systems, periodic solutions, numerical continuation, boundary value problems, three-body problem, figure-8 orbit. MSC: 34C25, 34C30, 34C14, 37J15, 65L10. 1 Introduction Our interest is about finding symmetric periodic solutions in the three body problem. In a previous paper [7] we studied the continuation of periodic orbits in the TBP following the scheme given in [13] for continuation in conservative systems. Unlike other methods, this scheme does not make use of symplectic reduction before numeric calculation. Several families of periodic orbits were shown in [7] and the numerical study of their linear stability (characteristic multipliers). Another applications of this technique appeared in [6] where 229