Every minimal non-degenerate critical Z/2 eigensection on S^2 is deformation rigid, so small changes to the branch-point configuration that preserve criticality must come from SO(3) rotations.
Parker.Z 2-harmonic spinors and 1-forms on connected sums and torus sums of 3-manifolds
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Deformation rigidity for Z/2 eigensections
Every minimal non-degenerate critical Z/2 eigensection on S^2 is deformation rigid, so small changes to the branch-point configuration that preserve criticality must come from SO(3) rotations.