manifold), then this is a singular hyperbolic set of $X$, in any dimension. Moreover, they proved that, $C^1$-generically in dimension four, the chain recurrent set of a star flow is singular hyperbolic.

In [@MPP04], it has been proved that every robustly transitive singular set for a three dimensional flow is a partially hyperbolic attractor or repeller and the singularities in this set must be Lorenz-like. In [@GanLiWen2005], Gan, Li and Wen extended the result in [@MPP04] to higher dimension assuming that the set is also strongly homogeneous. We recall that a vector field $X$ is said to be *strongly homogeneous of index $0 \leq {\operatorname{ind}}(\Lambda) \leq n-1$* over a set $\Lambda$ whether it cannot be $C^1$-approximated by one which has some hyperbolic periodic orbit of index different of ${\operatorname{ind}}(\Lambda)$ in a neighborhood $U$ of $\Lambda$. Here, the index ${\operatorname{ind}}(\cdot)$ means the dimension of the contracting subbundle from the hyperbolic splitting of a hyperbolic periodic orbit .

We recall that a compact invariant set $\Lambda$ is *robustly transitive* for a vector field $X$ if there exist a neighborhood $U$ of $\Lambda$ and a neighborhood ${{\mathcal U}}\in {\mathfrak{X}^{1}(M)}$ of $X$ such that, for every $Y \in {{\mathcal U}}$, the maximal invariant set 