Phase diagrams and excitations of anisotropic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> quantum magnets on the triangular lattice

The $S=1$ bilinear-biquadratic Heisenberg exchange model on the triangular lattice with a single-ion anisotropy has previously been shown to host number of exotic magnetic and nematic orders [Moreno-Cardoner $\textit{et al.}$, Phys. Rev. B $\textbf{90}$, 144409 (2014)], including an extensive region "supersolid" order. In this work, we amend by XXZ in interactions. Tuning limit exactly solvable generalized Ising-/Blume-Capel-type provides controlled access phases at finite transverse exchange. Notably, find additional macroscopically degenerate phase diagram study its fate under perturbation theory. We further map out diagrams as function parameter, ratio bilinear biquadratic interactions anisotropy, compute corrections total ordered moment various using systematically constructed linear flavor-wave also present spectra states, finding that lowest-energy band three-sublattice (i.e. $S^z=\pm1,0$) Ising/Blume-Capel stabilized strong anisotropies, is remarkably flat, opening up way flat-band engineering excitation via stabilizing non-trivial Ising-ordered ground states.