= Solution
Permuting the gaps, or equivalently relabelling the particles, leaves both quadratic forms invariant. The three-dimensional <permutation representation> splits as the rigid-rotation line spanned by $(1,1,1)$ and the two-dimensional sum-zero <standard representation of the symmetric group>.
The two displayed vibrational vectors merely choose a basis of that sum-zero plane. A permutation generally mixes them, but their common frequency leaves the whole plane invariant. This is precisely the <degenerate normal modes from permutation symmetry> required by the physical symmetry.
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