  We use spin wave theory to investigate the ground state properties of the
$Z_2$-invariant quantum XXZ model on the triangular lattice in the
ferromagnetic phase. The Hamiltonian comprises nearest and
next-nearest-neighbour Ising couplings, external magnetic fields, and a
$Z_2$-invariant ferromagnetic coupling. We show that quantum fluctuations are
suppressed in this system, hence linear spin wave theory gives reasonable
estimates of the ground state thermodynamic properties. Our results show that,
at half-filling (zero magnetic fields), the spontaneous breaking of $Z_2$
symmetry leads to a ferromagnetic phase whose energy spectrum is gapped at all
excitations with a maxon dispersion at $\mathbf{k}=0$. This is in sharp
contrast to rotational invariant systems with a vanishing phonon dispersion. We
show that the $\mathbf{k}=0$ mode enhances the estimated values of the
thermodynamic quantities. We obtain the trends of the particle density and the
condensate fraction. The density of states and the dynamical structure factors
exhibit fascinating peaks at unusual wave vectors, which should be of interest.
