\begin{document}$ \eta \bar{K}K^* $\end{document} three-body system in order to look for possible \begin{document}$ I^G(J^{PC}) = 0^+(1^{-+}) $\end{document} exotic states in the framework of the fixed-center approximation of the Faddeev equation. We assume the scattering of \begin{document}$ \eta $\end{document} on a clusterized system \begin{document}$ \bar{K}K^* $\end{document}, which is known to generate \begin{document}$ f_1(1285) $\end{document}, or a \begin{document}$ \bar{K} $\end{document} in a clusterized system \begin{document}$ \eta K^* $\end{document}, which is shown to generate \begin{document}$ K_1(1270) $\end{document}. In the case of \begin{document}$ \eta $\end{document}-\begin{document}$ (\bar{K}K^*)_{f_1(1285)} $\end{document} scattering, we find evidence of a bound state \begin{document}$ I^G(J^{PC}) = 0^+(1^{-+}) $\end{document} below the \begin{document}$ \eta{f_1(1285)} $\end{document} threshold with a mass of around 1700 MeV and a width of about 180 MeV. Considering \begin{document}$ \bar{K} $\end{document}-\begin{document}$ (\eta K^*)_{K_1(1270)} $\end{document} scattering, we obtain a bound state \begin{document}$ I(J^{P}) = 0(1^{-}) $\end{document} just below the \begin{document}$ \bar{K}{K_1(1270)} $\end{document} threshold with a mass of around 1680 MeV and a width of about 160 MeV."> Prediction of possible exotic states in the <inline-formula><tex-math id="M1">\begin{document}${\eta \bar{K}K^*}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/df927e27-716a-4c96-8b57-9e5740be7a5c/CPC-2019-0382_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/df927e27-716a-4c96-8b57-9e5740be7a5c/CPC-2019-0382_M1.png"/></alternatives></inline-formula> system -
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