We introduce a class of vector fields on n-manifolds containing the hyperbolic systems, the singular-hyperbolic systems on 3-manifolds, the multidimensional Lorenz attractors and the robust transitive singular sets in Li et al [Robust transitive singular sets via approach of an extended linear Poincar flow. Discrete Contin. Dyn. Syst. 13(2) (2005), 239269]. We prove that the closed orbits of a system in such a class are hyperbolic in a persistent way, a property which is false for higher-dimensional singular-hyperbolic systems. We also prove that the singularities in the robust transitive sets in Li et al are similar to those in the multidimensional Lorenz attractor. Our results will give a partial negative answer to Problem9.26 in Bonatti et al [Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III). Springer, Berlin, 2005]. © 2008 Cambridge University Press.