on the preperiodic set $P_*(X)$, $C^1$ along the flow direction over each preperiodic orbit, such that $X$ is strictly ${\EuScript{J}}$-separated on every $p \in P_*(X)$;

2.  for every $\sigma \in {\mathrm{Sing}}(X\vert_U)$ and $\forall v \in T_\sigma M, {\EuScript{J}}'(v) > 0$;

3.  the linear Poincaré flow $P^t$ associated to each preperiodic orbit $\gamma$ of $X\vert_U$ is strictly ${\EuScript{J}}$-monotone.

The definitions concerning the quadratic forms are given in the next section.

Let us explain why we are not requiring strict monotonicity in a whole nonsingular neighborhood of $X$. In fact, under certain assumptions, if we have strict monotonicity over any compact invariant nonsingular subset then it is a hyperbolic subset (see [@ArSal2012 Theorem D]). The idea here is to extend the characterization for any star flow, since it is already known the existence of star flows which are not neither uniformly hyperbolic nor singular/sectional hyperbolic (see [@BodL17]).

Now, we present the definition of strong homogeneity.

\[def:str-hom-set\] We say that a set $\Lambda$ is strongly homogeneous of index ${\operatorname{ind}}$ for a flow $X_t$, if there exist neighborhoods $U$ of $\Lambda$ and ${\EuScript{U}}$ of $X$ such that all periodic orbits in $U$ with respect to any flow in ${\EuScript{U}}$ have index ${\operatorname{ind}}$.

The second 