We present an analysis of the processes $e^{+}{e}^{-}\to \eta \mathrm{\Upsilon }(1S)$, $\eta \mathrm{\Upsilon }(2S)$, and $\gamma X_b$ with $X_b\to {\pi }^{+}{\pi }^{-}{\chi }_{bJ},{\chi }_{bJ}\to \gamma \mathrm{\Upsilon }(1S)$ ($J=1$, 2) reconstructed from $\gamma \gamma {\pi }^{+}{\pi }^{-}{\ell }^{+}{\ell }^{-}$ $(\ell =e,\mu )$ final states in $19.6{\mathrm{fb}}^{-1}$ of Belle II data collected at four energy points near the peak of the $\mathrm{\Upsilon }(10753)$ resonance. Here, $X_b$ is a hypothetical bottomonium-sector partner of the $X(3872)$. In the process $e^{+}{e}^{-}\to \eta \mathrm{\Upsilon }(2S)$, we observe a signal with a significance that exceeds $6.0\sigma$ when all energy points are combined. A signal with a significance of $4.2\sigma$ is observed at a center-of-mass energy of 10.653 GeV, close to the $B^{*}{\overline{B}}^{*}$ threshold. At this energy, we measure $\sigma (\eta \mathrm{\Upsilon }(2S))/\sigma ({\pi }^{+}{\pi }^{-}\mathrm{\Upsilon }(2S))>23$ at the 90% confidence level, indicating a strong violation of heavy-quark spin symmetry. No significant signal is observed for $e^{+}{e}^{-}\to \eta \mathrm{\Upsilon }(1S)$ or $\gamma X_b$. Upper limits on the Born cross sections for the processes $e^{+}{e}^{-}\to \eta \mathrm{\Upsilon }(1S)$ and $e^{+}{e}^{-}\to \gamma X_b$ with $X_b\to {\pi }^{+}{\pi }^{-}{\chi }_{bJ}$ are determined.

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• Hadronic decays
• Multiquark bound states
• Quantum chromodynamics