Using data collected by the Belle II experiment during 2018 at the SuperKEKB collider, we perform the first searches for the invisible decay of a $Z^prime$ in the process $e^+ e^- o mu^+ mu^- Z^prime$ and of a lepton-flavor-violating $Z^prime$ in $e^+ e^- o e^pm mu^mp Z^prime$. We do not find any excess of events and set 90$%$ credibility level upper limits on the cross sections of these processes. We translate the former, in the framework of an $L_mu-L_ au$ theory, into upper limits on the $Z^prime$ coupling constant at the level of $5 imes 10^-2div1$ for $M_Z^primeleq 6$ GeV/$c^2$.
