Isometry of a cone

Every isometry of a complete singular flat cone, of cone angle $\theta \ne 2\pi$, fixes its vertex: this is the unique point without a locally Euclidean neighbourhood. It preserves distance from that vertex and has the formThese are angular rotations and reflections. The stabilizer of a nonvertex point consists of the identity and its radial reflection. The vertex stabilizer is the entire isometry group, including all rotations: an unspecified fixed point does not give the two-element conclusion.

When $\theta =2\pi$, the complete cone is the Euclidean plane, allowing translations and every rotation about a fixed point. This smooth case is excluded from the singular-cone statement.