star flows satisfy Axiom A and no cycle condition. But, in the singular setting, this is no longer true. One of the most emblematic example that a star flow does not satisfy Axiom A is the geometric Lorenz Attractor [@Guck76; @Lo63].

From this, remained the question about singular star vector fields.

\[ex:geomLorenz\] The geometric Lorenz attractor is an emblematic example of star flow. see Figure \[fig:Lorenz\].

![The geometric Lorenz flow.[]{data-label="fig:Lorenz"}](L3D){width="6cm"}

In an attempt to study the behaviour of robust singular attractors like Lorenz ones, Morales, Pacífico and Pujals in [@MPP99] defined the so called *singular hyperbolic systems*.

Many researchers have worked about this notion in order to understand it as an extension of the hyperbolic theory for invariant sets for flows which are not (uniformly) hyperbolic, but which have some robust properties, certain kind of weaker hyperbolicity and also admit singularities.

The authors in [@MPP99] proved that, $C^1$-generically in dimension three, chain recurrence classes are singular hyperbolic. Without the generic assumption this result does not hold, as we can see in [@BaMo14].

In [@shi-gan-wen2014], Gan, Shi and Wen, proved that if a chain recurrent class of a star flow $X$ has homogeneous index for singularities (the same dimension of stable 