  The shear Alfv\'en continuum spectrum is studied for a tokamak with a single
island chain using the ideal Magnetohydrodynamics (MHD) theory. We have taken
into account the toroidal geometry and toroidal mode coupling with the island
considered as a highly-shaped stellarator. Various new frequency gaps open up
inside the island due to its asymmetry both poloidally and toroidally, such as
the Mirror-induced Alfv\'en Eigenmode (MAE) gap and the Helicity-induced
Alfv\'en Eigenmode (HAE) gap. We have shown that the MAE gap acts as the
continuation of the outside Toroidal Alfv\'en Eigenmode (TAE) gap into the
island. However, the combined TAE/MAE gap is getting narrower as the island
grows, leaving only half of its original width with a moderate island size as
much as 3.2% of the minor radius. In addition, the two-dimensional
eigenfunction of the continuum mode on the lower tip of the MAE gap now has
highly localised structures around the island's long axis, contrary to the
usual oscillatory global solutions found with no or a low level of toroidal
asymmetry - an indication of the continuous spectrum becoming discrete and
dense. These results have implications for the frequency, mode structure and
continuum damping of global TAEs residing in the gap.
