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We study a XXZ spin-chain in a gapless Tomonaga-Luttinger liquid (TLL) phasewith time dependent anisotropy of spin exchange interactions. To begin we focuson a linear ramp of $J_z$, starting at XX point and slowly increasing towardsthe anti-ferromagnetic Heisenberg point. Although the problem of a linear rampin the TLL has been recently under intense scrutiny in a perturbative\emph{g-ology} framework, an aspect that has been overlooked so far is the roleof the Galilean invariance breaking. We find that, although the differentialequation that needs to be solved to find time evolution of the system issubstantially more complicated, in some cases exact analytic solutions can begiven. We obtain them for the linear ramp in the limit of small $J_z$ as wellas $J_z\rightarrow 1$, and for such protocols that are tailored to keep theGalilean invariance breaking term constant for every $J_z$. We point out thefeatures of dynamics during the quench that stays unaltered, and those thatneed to be taken with care when Galilean invariance breaking is present. We areable to deduce that it is the shape of the propagating front that is affectedin the most pronounced way.

Slow quenches in XXZ spin-chains -- the role of Galilean invariance breaking