Maximal transitive sets with singularities for generic C1 vector fields

A transitive set Lambda of a vector field X is maximal transitive if it contains every transitive set of X intersecting it. We shall prove that if X is C-1 generic then every singularity of X with either only one positive or only one negative eigenvalue belongs to a maximal transitive set of X. In particular, we characterize maximal transitive sets with singularities for generic C-1 vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in [3] and [15].