  An integrable model perturbed by special ''weak integrability-breaking''
perturbations thermalizes at timescales much longer than predicted by Fermi's
golden rule. Recently, a systematic construction of such perturbations based on
the so-called long-range deformations of integrable chains was formulated.
These perturbations, obtained as truncations of the long-range deformations in
some small parameter expansions, can be viewed as produced by unitary rotations
of the short-range integrable models. For infinite systems, several
''generators'' (extensive local, boosted, and bilocal operators) of weak
perturbations are known. The main aim of this work is to understand the
appropriate generators in finite systems with periodic boundaries since simple
counterparts to boosted and bilocal operators are not known in such cases. We
approach this by studying the structure of the adiabatic gauge potential (AGP),
a proxy for such generators in finite chains, which was originally introduced
as a very sensitive measure of quantum chaos. We prove an exact relation
between the AGPs for the boosted and bilocal classes of generators and note
that the counterpart to boost does not seem to have a closed analytic form in
finite systems but shows quasi-locality nonetheless. We also introduce and
study strictly local variants of weak integrability-breaking perturbations.
