  We study smooth varieties of Picard number one admitting a special dominating
family of rational curves and an equalized $\mathbb{C}^*$-action. In particular
we show that $X$ is a smooth variety of Picard number one with nef tangent
bundle admitting an equalized $\mathbb{C}^*$-action with an isolated extremal
fixed point if and only if $X$ is an irreducible Hermitian symmetric space.
