spatial distribution of multiple stellar populations of $\omega\,$Centauri.

We emphasize that proper motions from Bellini and collaborators are not included in our study on the kinematics of $\omega\,$Centauri. Instead, as we will widely discuss in the next sections, the internal kinematics of multiple stellar populations in $\omega\,$Centauri and M22 are investigated by using high-precision proper motions from [*HST*]{} images and from Gaia DR2 alone.

[*HST*]{} dataset
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[*HST*]{} photometry and relative proper motions are used to investigate stellar populations of M22 and $\omega\,$Centauri with radial distances smaller than $\sim$1.7 and $\sim$2.5 arcmin, respectively.

To identify the stellar populations along the RGB of M22 and $\omega\,$Centauri, we used the catalogues by @milone2017 and @milone2018, which include photometry collected through the F275W, F336W, F438W and F814W bands of the Ultraviolet and Visual Channel of the Wide Field Camera 3 (UVIS/WFC3).

The main properties of the images that we used to derive relative stellar proper motions are summarized in Table \[tab:data\]. To derive the photometry and the astrometry of all the stars we used the FORTRAN software package KS2 developed by Jay Anderson, [see, e.g. @anderson2008; @sabbi2016 for details]. Since we are interested in proper motion determination, we reduced the images collected in 