On the singular hyperbolicity of star flows

We prove for a generic star vector field $X$ that, if for every chain recurrent class $C$ of $X$ all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular hyperbolic. We also prove that every Lyapunov stable chain recurrent class of $X$ is singular hyperbolic. As a corollary, we prove that the chain recurrent set of a generic 4-dimensional star flow is singular hyperbolic.