D].

Note that in the last result monotonicity is only required on preperiodic orbits.

If we require it over any nonsingular compact invariant subsets in $\Lambda$, it is possible to evaluate the index of a singularity, once it is accumulated by regular orbits.

\[mcor:strong-homog\] Let $\Lambda$ be a maximal invariant set of a neighborhood $U$ for $X \in {\mathfrak{X}^{1}(M)}$. Then, $\Lambda$ is strongly homogeneous with index ${\operatorname{ind}}$ and ${\operatorname{ind}}(\sigma) \geq {\operatorname{ind}}$ for all $\sigma \in {\mathrm{Sing}}(X\vert_{\Lambda})$ if there is a field of non-degenerate $C^1$ quadratic forms ${\EuScript{J}}$ on $U$ with index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}$ such that $X$ is strictly ${\EuScript{J}}$-separated, the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on every compact invariant nonsingular subset $K$ of $U$ and for every $\sigma \in {\mathrm{Sing}}(X\vert_U)$ and $\forall v \in T_\sigma M, {\EuScript{J}}'(v) > 0$.

We may ask if the converse of Corollary \[mcor:strong-homog\] is valid. But, just by supposing strongly homogeneity, we could not obtain the field of quadratic forms, because we need some kind of decomposition on the tangent bundle to create the cones.

\[mthm:strong-homog-to-quad\] Let $\Lambda$ be a compact invariant set whose singularities are hyperbolic (if any) and accumulated by regular orbits for a $C^1$ vector field $X$, which 