motions for Fe-poor and Fe-rich stars in each sector, and then subtracted the cluster median motion. As previously done for the radial and tangential velocity profiles, we account for Gaia systematic errors as in @vasiliev2019.

The resulting quantities, $\Delta\mu_{\rm \alpha}\cos\delta$ and $\Delta\mu_{\rm \delta}$, are shown in the bottom panels of Figure \[fig:rot\] as a function of the position angle $\theta$, defined counter-clockwise from the east. A visual inspection of this figure suggests that, consistently with the previous method, both populations of $\omega\,$Centauri and M22 rotate in the plane of the sky. This result is illustrated in the top panels of Figure \[fig:rot\], where we show the positions of the stars relative to the cluster center, together with the vectors that indicate the median motions of Fe-poor and Fe-rich stars computed in each circular sector.

As in [@cordoni2019], we used the least-squares technique to fit the sine function $$\label{eq:1}
f(\theta)=M+A\cdot\sin(F\cdot\theta+\phi)$$ to all Fe-poor and Fe-rich stars in $\omega\,$Centauri and M22. The best fit functions to Fe-poor and Fe-rich stars are represented with orange and azure lines, respectively, in the bottom panels of Figure \[fig:rot\]. We estimate the goodness of the fit by means of the $r^2$ parameter: $$\label{eq:2}
r^2=1-\frac{\sum_i  (y_i-f(\theta,i))^2}{\sum_i(f (\theta, 