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17.1 Groups G 2 on non-null orbits 267Then (17.10) shows that for ε = −1 we obtain the general solution√ √W = f(u)+g(v), 2u = t − z, 2v = t + z, (17.12)with arbitrary functions f(u) and g(v), and for ε =+1W is analytic.In the vacuum case with A = 0 in (17.4) (diagonal form of the metric),the function Ψ obeys the linear differential equation(W Ψ ,3 ) ,3 + ε(W Ψ ,4 ) ,4 = 0 (17.13)(see also §25.2), and the rest of the field equations determine M in termsof a line integral.In most applications (e.g. for stationary axisymmetric gravitationalfields or colliding plane waves) the condition (17.3) for orthogonal transitivityfollows from other physical assumptions. The more general casewhen the metric with two commuting Killing vectors is not reducible toblock-diagonal form has been investigated by Gaffet (1990). In that paperthe field equations and a corresponding Lagrangian were derived, reductionsgave rise to several cases of integrability, either by quadratures, orby elliptic functions, and a non-orthogonally transitive generalization ofWeyl’s static metrics (§20.2) was obtained.Non-orthogonally transitive metrics with a preferred null direction(e.g. a pure radiation source) have been discussed by Kolassis andSantos (1987).17.1.3 G 2 II on non-null orbitsThe case of the non-Abelian group G 2 II has been considered in the literatureonly occasionally. A reason for that may be that the Lorentz groupdoes not possess a G 2 II as a subgroup so that a G 2 II will not occurwithin asymptotically flat solutions, in agreement with the result derivedby Carter (1970).Kolassis (1989) formulated the necessary and sufficient conditions for aspace-time to admit a G 2 I or a G 2 II in the context of the modified spincoefficient formalism of Geroch et al. (1973) (§7.4).Aliev and Leznov (1992a, 1992b) have rewritten Einstein’s vacuumequations in the form of a covariant gauge theory in two dimensions.A special solution of the field equations is the metric AII in Table 18.2(all metrics with pseudospherical symmetry admit a G 2 II, cp. §8.6.2); asecond ansatz reduces the problem to an ordinary differential equation,and a third ansatz leads to a solution with an additional timelike Killingvector.An example of an algebraically special vacuum solution is provided by(38.6). Perfect fluid solutions with a G 2 II are considered in Chapter 23.