What does "unremarkable transverse peculiar velocity" mean exactly, and how is it calculated here? The abstract of A magnetar parallax (also in MNRAS) contains the following:

Combining our new observations with two archival observations from 2006, we have refined the proper motion and reference position of the magnetar and have measured its annual geometric parallax, the first such measurement for a magnetar. The parallax of 0.40±0.05mas corresponds to a most probable distance 2.5+0.4−0.3kpc for J1810. Our new astrometric results confirm an unremarkable transverse peculiar velocity of ≈200km s−1 for J1810, which is only at the average level among the pulsar population

This reports the first radio astrometric determination of parallax &quot;for a magnetar&quot;.
The peculiar phrase &quot;unremarkable transverse peculiar velocity&quot; is vernacularly dissonant, but parenthetical clarifications in the body of the paper help with that:

5 DISCUSSION
As is shown in Table 1, our new proper motion significantly improves on the previous value inferred from the two year-2006 positions; the new distance D = 2.5+0.4−0.3 kpc is consistent with 3.1 ± 0.5 kpc estimated using red clump stars (Durant &amp; van Kerkwijk 2006), while in mild tension with 3.1−4.0 kpc constrained with neutral-hydrogen absorption (Minter et al. 2008), suggesting the  distance to the neutral-hydrogen screen was over-estimated. In models of NS kicks from the electromagnetic rocket effect (Harrison &amp; Tademaru 1975) one might expect magnetars to have higher velocities (Duncan &amp; Thompson 1992). Our new parallax and proper motion corresponds to the transverse velocity vt = 198+29−23 km s−1. Using the Galactic geometric parameters provided by Reid et al. (2019) and assuming a flat rotation curve between J1810 and the Sun, the peculiar velocity (with respect to the neighbourhood of J1810) perpendicular to the line of sight was calculated to be vb = −54 ± 8 km s−1 and vl = −175 ± 26 km s−1. Our refined astrometric results consolidate the conclusion by Helfand et al. (2007) that J1810 has a peculiar velocity typically seen in “normal” pulsars, unless its radial velocity is several times larger than the transverse velocity.

My understanding of that is limited, but I think that &quot;peculiar velocity&quot; is the velocity relative to some larger structure in which it is found and/or believed to be gravitationally associated with, and the &quot;perpendicular peculiar velocity&quot; component comes from some model that includes the assumption of &quot;a flat rotation curve between J1810 and the Sun&quot;. But I haven't a clue what that means.
See also Wikipedia's Peculiar_velocity; Galactic_astronomy for a short description.

In galactic astronomy, peculiar motion refers to the motion of an object (usually a star) relative to a Galactic rest frame.

Questions:

How is the transverse component of said peculiar velocity determined here, and what does &quot;a flat rotation curve between J1810 and the Sun&quot; refer to?
Have I got this right, and in general &quot;peculiar velocity&quot; is a velocity relative to some larger structure in which it is found and/or believed to be gravitationally associated with?
&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;