  Let $X$ be a compact strongly pseudoconvex CR manifold with a transversal CR
$S^1$-action. In this paper, we establish the asymptotic expansion of Szeg\H{o}
kernels of positive Fourier components and by using the asymptotics, we show
that $X$ can be equivariant CR embedded into some $\mathbb C^N$ equipped with a
simple $S^1$-action. An equivariant embedding of quasi-regular Sasakian
manifold is also derived.
