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Theories beyond the standard model often predict the existence of anadditional neutral boson, the $Z^{\prime}$. Using data collected by the BelleII experiment during 2018 at the SuperKEKB collider, we perform the firstsearches for the invisible decay of a $Z^{\prime}$ in the process $e^+ e^- \to\mu^+ \mu^- Z^{\prime}$ and of a lepton-flavor-violating $Z^{\prime}$ in $e^+e^- \to e^{\pm} \mu^{\mp} Z^{\prime}$. We do not find any excess of events andset 90\% credibility level upper limits on the cross sections of theseprocesses. We translate the former, in the framework of an $L_{\mu}-L_{\tau}$theory, into upper limits on the $Z^{\prime}$ coupling constant at the level of$5 \times 10^{-2}$ -- $1$ $M_{Z^\prime}\leq 6$ GeV/$c^2$.

Search for an invisibly decaying $Z^{\prime}$ boson at Belle II in $e^+ e^- \to \mu^+ \mu^- (e^{\pm} \mu^{\mp})$ plus missing energy final states