  We investigate the nature of a $Z_2$-invariant XY ring-exchange interaction
with a frustrated Ising coupling on the triangular lattice. In the limit of
pure XY ring-exchange interaction, we show that the classical ground state is
degenerate resulting from the $Z_2$-invariance of the Hamiltonian. Quantum
fluctuations lift these classical degenerate ground states and produce an
unusual state whose excitation spectrum exhibits a gapped maximum quadratic
dispersion near ${\bf k}=0$ and vanishes at the midpoints of each side of the
Brillouin zone. This result is in contrast to a gapless quadratic dispersion
near ${\bf k}=0$ in the U(1)-invariant counterpart. We also study the effects
of frustration when competing with a classically frustrated Ising interaction.
We provide a glimpse into the possible quantum phases that could emerge. A
comprehensive understanding of this Hamiltonian, however, cannot be elucidated
analytically and requires an explicit numerical simulation.
