Birkhoff's Theorem for Three-Dimensional AdS Gravity
  All three-dimensional matter-free spacetimes with negative cosmological constant, compatible with cyclic symmetry are identified. The only cyclic solutions are the 2+1 (BTZ) black hole with SO(2) x R isometry, and the self-dual Coussaert-Henneaux spacetimes, with isometry groups SO(2) x SO(2,1) or SO(2) x SO(2). 
Eloy Ay\'on-Beato, Cristi\'an Mart\'inez and Jorge Zanelli