$\Lambda_Y = \cap_{t \in {{\mathbb R}}} X_t(U)$ is contained in the interior of $U$ and is non-trivially (not a single orbit) transitive.

Many researchers believed that, at least generically, singular star flows should be singular hyperbolic ones. There are many conjectures about this and various results involving the most varied assumptions, see for instance, [@ArbMo2013; @GanLiWen2005; @shi-gan-wen2014].

Another point of view on the influence of the singularities emerged in the recent work of Bonatti and da Luz [@BodL17], where the authors have defined *multisingular hyperbolicity* which admit singularities of different indexes in the same recurrent class, and have shown that this kind of hyperbolicity implies star condition. It is easy to see that the classical singular hyperbolic flows (central dimension $2$ or the full dimension of central subbundle) are a particular case of the multisingular ones. Moreover, they proved that the converse holds on the $C^1$ generic assumption.

In this paper, we prove a characterization of star condition for flows via ${\EuScript{J}}$-algebra of Potapov (infinitesimal Lyapunov functions) [@Pota60; @Pota79; @Wojtk01]. We also show a relation between ${\EuScript{J}}$-algebra and strong homogeneity. Then, we apply this to obtain some results about singular hyperbolic systems.

Recall that a pseudo-euclidean space is a real 