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In this paper, the boundedness of a large class of sublinear commutator operators $T_b$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_{p,\varphi }\left(w\right)$ with the weight function $w$ belonging to Muckenhoupt’s class $A_p$ is studied. When $1<p<\infty$ and $b\in \mathrm{BMO}$, sufficient conditions on the pair $({\varphi }_1,{\varphi }_2)$ which ensure the boundedness of the operator $T_b$ from $M_{p,{\varphi }_1}\left(w\right)$ to $M_{p,{\varphi }_2}\left(w\right)$ are found. In all cases the conditions for the boundedness of $T_b$ are given in terms of Zygmund-type integral inequalities on $({\varphi }_1,{\varphi }_2)$, which do not require...