We study vacuum solutions of three-dimensional f ( Q ) gravity exhibiting functional under determination, where the field equations fail to uniquely fix all metric components despite imposed symmetries. For static, circularly symmetric spacetimes, we identify a special nonanalytic model f ( Q ) ∝ Q in which the reduced equations become degenerate, admitting an infinite-dimensional family of black-hole exterior solutions. While the lapse function, horizon radius, and asymptotic AdS 3 structure are universal and coincide with BTZ-like geometries, one radial metric component remains arbitrary, yielding locally inequivalent bulk geometries without affecting the horizon or asymptotics. We formulate a consistent variational principle, compute the conserved mass via the renormalized Brown–York stress tensor, and show that no additional asymptotic charges arise. A Hamiltonian analysis demonstrates that the functional freedom originates from a rank-deficient constraint algebra and an extra first-class constraint, identifying it as a gauge redundancy rather than new propagating degrees of freedom. Thermodynamically, the mass and horizon radius are universal, whereas the Hawking temperature and entropy depend on the horizon value of the undetermined function, with the first law holding for variations at fixed gauge profile. The geodesic structure exhibits a clear separation between universal features governed by the lapse function and function-dependent effects affecting local trajectories, implying a universal black-hole shadow boundary with modified photon scattering outside it. Finally, a topological analysis based on the generalized off-shell free energy shows that the solutions correspond to a single thermodynamic defect with unit winding number.
Nashed et al. (Sun,) studied this question.