= Solution
Use polar coordinates $z_j=r_je^{i\theta_j}$ on each complex coordinate plane. On $S^3$, where $r_1^2+r_2^2=1$,
$$
\alpha_s=\frac12(r_1^2d\theta_1+sr_2^2d\theta_2),
\qquad
d\alpha_s=r_1dr_1\wedge d\theta_1+sr_2dr_2\wedge d\theta_2.
$$
Direct substitution shows that $\alpha_s\wedge d\alpha_s$ is nowhere zero for $s>0$, so $\alpha_s$ is a <contact form>.
The vector field
$$
R_s=2\frac{\partial}{\partial\theta_1}+\frac2s\frac{\partial}{\partial\theta_2}
$$
satisfies $\alpha_s(R_s)=r_1^2+r_2^2=1$ and $\iota_{R_s}d\alpha_s=0$ on tangent vectors to the sphere. It is therefore the <Reeb vector field>. Its flow is
$$
(z_1,z_2)\longmapsto(e^{2it}z_1,e^{2it/s}z_2).
$$
If both coordinates are nonzero, an orbit closes only if the two angular frequencies have rational ratio, equivalently if $s\in\mathbb Q$. For irrational $s$, the only closed orbits are $\{z_2=0\}$ and $\{z_1=0\}$, the two coordinate circles. This is the <irrational contact ellipsoid flow on the three-sphere>.
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