  The calculation of the parity-violating polarizations in the external
electric field, which are associated with the electron electric dipole moment
($e$EDM) and magnetic quadrupole moment (MQM) of the $^{175}$Lu nucleus, as
well as the determination of the rovibrational structure for the
$^{175}$LuOH$^+$ cation, is performed. Beyond the bending of the molecule, the
slight effect of the stretching of the distance between Lu and OH is taken into
account. This study is required for the preparation of the experiment and for
the extraction of the $e$EDM and MQM values of $^{175}$Lu from future
measurements.
