Motivated by recent quantum Monte Carlo (QMC) simulations of the quantum Kagome ice model by Juan Carrasquilla, et al., [Nature Communications 6, 7421 (2015)], we study the ground state properties of this model on the triangular lattice. In the presence of a magnetic field $h$, the Hamiltonian possesses competing interactions between a $Z_2$-invariant easy-axis ferromagnetic interaction $J_{\pm\pm}$ and a frustrated Ising term $J_z$. As in the U(1)-invariant model, we obtain four classical distinctive phases, however, the classical phases in the $Z_2$-invariant model are different. They are as follows: a fully polarized (FP) ferromagnet for large $h$, an easy-axis canted ferromagnet (CFM) with broken $Z_2$ symmetry for small $h$ and dominant $J_{\pm\pm}$, a {\it ferrosolid} phase with broken translational and $Z_2$ symmetries for small $h$ and dominant $J_{z}$, and two lobes with $m=\langle S_z\rangle=\pm 1/6$ for small $h$ and dominant $J_{z}$. We show that quantum fluctuations are suppressed in this model, hence the large-$S$ expansion gives an accurate picture of the ground state properties. When quantum fluctuations are introduced, we show that the {\it ferrosolid} state is the ground state in the dominant Ising limit at zero magnetic field. It remains robust for $J_z\to\infty$. With nonzero magnetic field the classical lobes acquire a finite magnetic susceptibility with no $S_z$-order. We present the trends of the ground state energy and the magnetizations. We also present a detail analysis of the CFM.

13 pages with 19 figures