is strongly homogeneous with index ${\operatorname{ind}}$ and ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$ for all $\sigma \in {\mathrm{Sing}}(X\vert_{\Lambda})$. Then, there exists a field of non-degenerate quadratic forms ${\EuScript{J}}$ on $\Lambda$ with index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}(\Lambda)$ for which $X$ is ${\EuScript{J}}$-separated and the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on every compact invariant nonsingular subset $\Gamma$ of $\Lambda$.

As an application, we obtain some results about partial hyperbolicity for robustly transitive strongly homogeneous singular sets of [@GanLiWen2005] and transitive set of [@ArbMo2013].

The text is organized as follow. In the first section, it is given the main definitions and stated the main results. In second section, it is presented the main tools by using the notion of ${\EuScript{J}}$-algebra of Potapov. In third section, it is given some applications concerning singular hyperbolicity. In fourth section is proved the main theorems.

Some definitions and auxiliary results {#sec:prelim-definit}
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Now, we give some definitions.

Let $M$ be a connected compact finite $d$-dimensional manifold, $d \geq 3$, without boundary, together with a flow $X_t : M \to M, t \in \mathbb{R}$ generated by a $C^1$ vector field $X: M \to TM$.

An *invariant set* $\Lambda$ for the flow of $X$ is a subset of $M$ which 