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 isomet... View more
Degenerate case: c33 = 0 If the fields KF1,T1 and KF2,T2 are null (c11 = 0) and spacelike (c22 > 0) respectively, they span a null plane (case III above). In this case c33 = 0, and therefore KF1,T1 , KF2,T2 and [KF1,T1 , KF2,T2 ] are not linearly independent. However, it is possible to find another independent null Killing vector which, together with KF1,T1 and KF2,T2 , generate the same so(2, 1) algebra. Without loss of generality they KF1,T1 and KF2,T2 can be taken to be orthogonal (c12 = 0), and their commutator is not linearly independent, but is given by
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