Rossi's non-embeddable CR structure

Counterexample

Let $S^3 = \set{(z_1, z_2) \in \C^2 : \abs{z_1}^2 + \abs{z_2}^2 = 1}$ be the unit sphere. Its standard CR structure is spanned by the vector field $$\bar L = z_2 \frac{\partial}{\partial \bar z_1} - z_1 \frac{\partial}{\partial \bar z_2},$$ which is tangent to $S^3$, and its conjugate is $L = \bar z_2 \frac{\partial}{\partial z_1} - \bar z_1 \frac{\partial}{\partial z_2}$. For a real number $t$ with $0 < \abs{t} < 1$, let $\bar L_t = \bar L + t L$, and let $S^3_t$ be $S^3$ with the CR structure spanned by $\bar L_t$. Then $S^3_t$ is a compact strongly pseudoconvex CR manifold of real dimension $3$. Every CR function $f$ on $S^3_t$, that is every $f$ with $\bar L_t f = 0$, is even: $f(-p) = f(p)$ for all $p \in S^3$. So CR functions cannot separate the antipodal points $p$ and $-p$, and there is no CR embedding of $S^3_t$ into any $\C^N$.