Question regarding the Milky Way when calculating galactic space velocities for galaxies
v$_r$ and μ are radial velocities and proper motion as seen from the Sun (ie heliocentric)  and the equations in the appendix convert to v in the $S_o$ frame which is the center of the Galaxy.  So, the motion of the MW is (0,0,0).  If you want to, you can solve for v$_r$ and μ for the MW by plugging v=(0,0,0) into the equation, and find the heliocentric proper motion and radial velocity of the Galactic center. But it is not needed.