  The triaxial projected shell model (TPSM) approach is generalized to
investigate the negative parity band structures in even-even systems. In the
earlier version of the TPSM approach, the quasiparticle excitations were
restricted to one major oscillator shell and it was possible to study only
positive parity states in even-even systems. In the present extension, the
excited quasiparticles are allowed to occupy two major oscillator shells, which
makes it possible to generate the negative parity states. As a major
application of this development, the extended approach is applied to elucidate
the negative parity high-spin band structures in $^{102-112}$Ru and it is shown
that energies obtained with neutron excitation are slightly lower than the
energies calculated with proton excitation. However, the calculated aligned
angular momentum ($i_x$) clearly separates the two spectra with neutron $i_x$
in reasonable agreement with the empirically evaluated $i_x$ from the
experimental data, whereas proton $i_x$ shows large deviations. Furthermore, we
have also deduced the transition quadrupole moments from the TPSM wavefunctions
along the negative-parity yrast- and yrare- bands and it is shown that these
quantities exhibit rapid changes in the bandcrossing region.
