  We consider a compact connected CR manifold with a transversal CR locally
free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove
that a certain weighted Fourier-Szeg\H{o} kernel of the CR sections in the high
tensor powers admits a full asymptotic expansion and we establish $\mathbb
R$-equivariant Kodaira embedding theorem for CR manifolds. Using similar
methods we also establish an analytic proof of an $\mathbb R$-equivariant
Boutet de Monvel embedding theorem for strongly pseudoconvex CR manifolds. In
particular, we obtain equivariant embedding theorems for irregular Sasakian
manifolds. As applications of our results, we obtain Torus equivariant Kodaira
and Boutet de Monvel embedding theorems for CR manifolds and Torus equivariant
Kodaira embedding theorem for complex manifolds.
