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268 17 Groups G 2 and G 1 on non-null orbitsPhan (1993) found a Petrov type I vacuum solution admitting a G 2 IIwith Killing vectors ∂ x and x∂ x − ∂ y and a homothetic vector ∂ y − ∂ t . Itreadsds 2 =2e 2y dx 2 + 8z [ (3 e−2t dy 2 + e−2t dzz 2zA + dy ) ]2− (dt − Hdy) 2 ,B√√A(z) = [(2z − 1) 2 +2]/8z, B(z) = 3/4z, (17.14)H(z) =1+2A(z)/B(z).Orthogonal transitivity and spacelike orbits of the group G 2 II reducethe metric (17.2) to the simpler formds 2 =e M (dz 2 − dt 2 )+W [e Ψ (e −y dx + Ady) 2 +e −Ψ dy 2 ], (17.15)where the metric functions M,Ψ,A and W (W > 0) depend only on zand t. The line element (17.15) differs from (17.4) only by the factor e −y .Bugalho (1987) has shown that the vacuum field equations imply that Ψand A in (17.2) are necessarily constants and that the resulting solutionsare pseudospherically-symmetric, i.e. they admit at least one additionalKilling vector. The same is true for perfect fluids for which µ+p >0 withthe four-velocity orthogonal to the group orbit.17.2 Boost-rotation-symmetric space-timesIn Minkowski space-time, the generators of rotations around the z-axisand of boosts along the z-axis commute; hence these two symmetriesform an Abelian group G 2 I of motions. Accordingly, curved space-timesare called boost-rotation-symmetric if they admit, in addition to thehypersurface-orthogonal Killing vector describing axial symmetry, asecond Killing vector which becomes a boost in the flat-space limit. Thecorresponding symmetries are compatible with gravitational radiationand asymptotic flatness. In axially-symmetric space-times, the onlyallowable second symmetry that does not exclude radiation is boostsymmetry. The proof (Bičák and Schmidt (1984), for corrections seeBičák and Pravdová (1998)) uses the asymptotic form of the Killingequations near future null infinity I + in Bondi coordinates.Asymptotically flat radiative space-times with boost-rotation symmetrywere systematically investigated by Bičák and Schmidt (1989) (thisreview article and the papers by Schmidt (1996) and Bičák (1997) mayalso be consulted for global aspects of the boost-rotation symmetry). Inaccordance with the metric (17.4) and the field equation W ,uv = 0, see(17.10), the canonical form of the metric considered by these authors in