What does "unremarkable transverse peculiar velocity" mean exactly, and how is it calculated here?
&quot;Peculiar velocity&quot; is a fixed term and describes the velocity of an object relative to a defined rest frame.
Astronomy has the problem that you need different methods to measure the 3D motion of an object. Therefor one often only gives the velocity within line-of-sight (from spectrographic data) or the perpendicular velocity as measured from astrometry, thus positional data on a sphere; that is the perpendicular velocity (to the line-of-sight).
As such the &quot;perpendicular peculiar velocity&quot; is the velocity of the magnetar as determined from astrometry - as also stated in your quote.
The reference frame of rest in this context is the expected motion of the stars around the galactic centre - at the distance of the observed magnetar. Galaxies don't have a simple Keplerian velocity where orbital speed reduces with distance to a central heavy central object like in our solar system. Dark matter changes the relationship between orbital distance and orbital speed in a way which is subject to investigation. With a &quot;flat rotation curve between J810 and the Sun&quot; they mean that orbital velocities don't differ, thus they indicate directly their normalization, their reference rest frame.
The peculiar velocity of a star within our Galaxy thus refers to their motion relative to their rest frame which orbits the Galactic center in the way expected for their distance from the center of the Galaxy.
So how was it measured? They say &quot;from annual parallax measurements&quot;. That means they measure the positon repeatedly and very exactly. This allows to measure the parallax in reference to the background. But at the same time you get the proper motion, thus the change of the positon when you removed the expected apparent change due to parallax. A typical astrometry series looks like  which I took from http://spiff.rit.edu/classes/phys240/lectures/parallax/parallax.html#nottrivial You nicely see the annual variation (the parallax) and the proper motion which remains when the circular annular motion is subtracted. This velocity is the perpendicular or transverse velocity.
Convert this measured velocity in the image plane to a real velocity by taking into account the distance to the object, reduce it by what you expect it to be for the Galactic distance of the object, and find that it is pretty normal to what you expect and you have an &quot;unremarkable transverse peculiar velocity&quot;