In this paper, we consider C1 vector fields X in R3 having a "generalized heteroclinic loop" L which is topologically homeomorphic to the union of a 2-dimensional sphere S2 and a diameter Γ connecting the north with the south pole. The north pole is an attractor on S2 and a repeller on Γ. The equator of the sphere is a periodic orbit unstable in the north hemisphere and stable in the south one. The full space is topologically homeomorphic to the closed ball having as boundary the sphere S2. We also assume that the flow of X is invariant under a topological straight line symmetry on the equator plane of the ball. For each n ∈ N, by means of a convenient Poincaré map, we prove the existence of infinitely many symmetric periodic orbits of X near L that gives n turns around L in a period. We also exhibit a class of polynomial vector fields of degree 4 in R3 satisfying this dynamics. © 2009 Elsevier B.V. All rights reserved.2386699705Lamb, J.S., Teixeira, M.A., Webster, K.N., Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R3 (2005) J. Differential Equations, 219, pp. 78-115Moser, J., (1973) Stable and Random Motions in Dynamical Systems, , Princeton University Press, Princeton New JerseySmale, S., Differentiable dynamical systems (1967) Bull. Amer. Math. Soc., 73, pp. 747-817Robinson, C., (1999) Studies in Advanced Mathematics, , CRC Press, Boca Raton, FLCorbera, M., Llibre, J., Generation of symmetric periodic orbits by a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2 in R3 (2007) Internat. J. Bifur. Chaos, 17, pp. 3245-3302Corbera, M., Llibre, J., Pérez-Chavela, E., Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2 (2006) J. Phys. A, 39, pp. 15313-15326Algaba, A., Fernández-Sánchez, F., Freire, E., Merino, M., Rodríguez-Luis, A.J., Nontransversal curves of T-points: A source of closed curves of global bifurcations (2002) Phys. Lett. A, 303, pp. 204-211Llibre, J., Ponce, E., Teruel, A., Horseshoes near homoclinic orbits for piecewise linear differential systems in R3 (2007) Internat. J. Bifur. Chaos, 17, pp. 1171-1184Shil'nikov, L.P., A case of the existence of a denumerable set of periodic motions (1965) Sov. Math. Dokl., 6, pp. 163-166Shil'nikov, L.P., A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle-focus type (1970) Math. USSR Sbornik, 10, pp. 91-102Shilnikov, L., Turaev, D.V., A new simple bifurcation of a periodic orbit of blue sky catastrophe type. Methods of qualitative theory of differential equations and related topics (2000) Amer. Math. Soc. Transl. Ser. 2, 200, pp. 165-188. , Amer. Math. Soc., Providence, RIShilnikov, A., Shilnikov, L., Turaev, D., Blue-sky catastrophe in singularly perturbed systems (2005) Mosc. Math. J., 5, pp. 269-282Burke, J., Knobloch, E., Homoclinic snaking: Structure and stability (2007) Chaos, 17, p. 037102. , 15 ppBurke, J., Knobloch, E., Localized states in the generalized Swift-Hohenberg equation (2006) Phys. Rev. E (3), 73, p. 056211. , 15 ppKnobloch, J., Wagenknecht, T., Homoclinic snaking near a heteroclinic cycle in reversible systems (2005) Physica D, 206, pp. 82-93Woods, P.D., Champneys, A.R., Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation (1999) Physica D, 129, pp. 147-170