= Solution
A common shift of all three <angles> leaves every spring length unchanged. This continuous rotational <symmetry> gives the <zero mode>: there is no restoring force against rigid rotation.
The equilateral <equilibrium> also has the <dihedral group> symmetry $D_3$. Its action on the three angular displacements preserves the uniform one-dimensional subspace and the two-dimensional sum-zero subspace. The latter is an <irreducible representation> over the <real numbers>; a reflection may reverse all angular signs as well as exchange bead labels, which does not change these invariant subspaces. The symmetric <stiffness matrix> must act as a scalar on the sum-zero subspace, explaining the equal vibrational <angular frequencies>. Directly, each sum-zero vector satisfies $(3I-\mathbf1\mathbf1^T)\mathbf v=3\mathbf v$. This is an example of <degenerate normal modes from permutation symmetry>, and of <normal modes of three equal masses with a symmetric gap potential>.
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