  We present a detailed theoretical analysis of a peculiar generation of
multiple bound states in the continuum (BICs) in two-dimensional periodic
arrays of dielectric spheres. They emerge in high-symmetry lattices with the
$C_{6v}$ and $C_{4v}$ point groups and involve doubly degenerate quasi-guided
modes at the $\Gamma$ point that can couple to external radiation. By tuning a
system parameter, the doubly degenerate modes can exhibit accidental BICs at a
critical parameter. In the vicinity of the critical parameter, the two bands
originating from the degenerate mode exhibit multiple off-$\Gamma$ BICs. They
move and annihilate across the two bands by changing the parameter around the
critical one. A ring-like high $Q$ channel pinned with multiple BICs emerges
particularly for the $C_{6v}$ case. Across the critical coupling, the total
vorticity of the multiple BICs is conserved. The $\vb*{k}\cdot\vb*{p}$
perturbation theory explains some features of the phenomena reasonably well.
