  We show that non-trivial chain recurrent classes for generic $C^1$ star flows
satisfy a dichotomy: either they have zero topological entropy, or they must be
isolated. Moreover, chain recurrent classes for generic star flows with zero
entropy must be sectional hyperbolic, and cannot be detected by any non-trivial
ergodic invariant probability. As a result, we show that $C^1$ generic star
flows have only finitely many Lyapunov stable chain recurrent classes.
