Observation of $e^+e^-\toηΥ(2S)$ and search for $e^+e^-\toηΥ(1S),~γX_b$ at $\sqrt{s}$ near 10.75 GeV

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