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17Groups G 2 and G 1 on non-null orbits17.1 Groups G 2 on non-null orbits17.1.1 Subdivisions of the groups G 2The groups G 2 can be divided into several subclasses depending on theproperties of the (appropriately chosen) two Killing vectors which (a) docommute or not, (b) are orthogonally transitive or not, (c) are hypersurface-orthogonalor not. We shall discuss these alternatives now in turn.(a) The normal forms of the space-time metrics in the commuting (G 2 I)and in the non-commuting case (G 2 II)(§8.2) are given by (Petrov 1966,p. 150)G 2 I : g ij = g ij (x 3 ,x 4 ), ξ = ∂ 1 , η = ∂ 2 , (17.1)⎛⎞e −2x2 a 11 e −x2 a 12 e −x2 a 13 0eG 2 II : g ij =−x2 a⎜ 12 a 22 a 23 0⎟⎝ e −x2 a 13 a 23 a 33 0 ⎠ , (17.2)0 0 0 e 4a ij = a ij (x 3 ,x 4 ), e 4 = ±1, ξ = ∂ 1 , η = x 1 ∂ 1 + ∂ 2 .The 2-surfaces of transitivity (group orbits) spanned by the two Killingvectors ξ and η are spacelike or timelike respectively when the square ofthe simple bivector ξ [a η b] is positive or negative.The general field equations, both for G 2 I and G 2 II, for space-timesadmitting a group G 2 of motions are very complicated to solve and noexact solutions have been obtained for either of the metrics (17.1), (17.2)without additional simplifications. The further restrictions imposed maybe degeneracy of the Weyl tensor, or special properties of the Killing vectorfields (see below), or an additional homothetic vector. The symmetry264