  We have performed a combined analysis of the BESIII data for both the
$Z_c(3900)$ and $Z_{cs}(3985)$ structures, assuming that the latter is an SU(3)
flavor partner of the former one. We have improved on the previous analysis of
Albaladejo $et$ $al.$ [Phys. Lett. B 755, 337 (2016)] by computing the
amplitude for the $D_1\bar{D}D^*$ triangle diagram considering both $D$ and
$S$-wave $D_1D^*\pi$ couplings. We have also investigated effects from SU(3)
light-flavor violations, which are found to be moderate and of the order of
20%. The successful reproduction of the BESIII spectra, in both the
hidden-charm and hidden-charm strange sectors, strongly supports that the
$Z_{cs}(3985)$ and $Z_c(3900)$ are SU(3) flavor partners placed in the same
octet multiplet. The best results are obtained when an energy-dependent term in
the diagonal $D^{(*)}\bar D_{(s)}^{(*)}$ interaction is included, leading to
resonances (poles above the thresholds) to describe these exotic states. We
have also made predictions for the isovector $Z_{c}^*$ and isodoublet
$Z_{cs}^*$, $D^*\bar{D}^*$ and $D^*\bar{D}_{s}^*$ molecules, with
$J^{PC}=1^{+-}$ and $J^{P}=1^{+}$, respectively. These states would be
heavy-quark spin symmetry partners of the $Z_{c}$ and $Z_{cs}$. Besides the
determination of the masses and widths of the $Z_c(3900)$ and $Z_{cs}(3985)$,
we also predict those of the $Z_c^*$ and $Z_{cs}^*$ resonances.
