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19.2 Orthogonal surfaces 293coordinates q m and x m respectively), the expansionsη m =(x n − q n )∇ n η m | q + O(2),X ≡ η m η m =(x m − q m )(x n − q n )H mn | q + O(3)(19.2)hold and therefore the regularity conditionX ,a X ,a /4X → 1 (19.3)in the limit at the rotation axis is satisfied. Otherwise, if (19.3) is violated,there are conical singularities (rods or struts) on the axis.The behaviour of the metric near the axis was carefully studied byCarlson and Safko (1980) using pseudo-Cartesian coordinates and theirtransformation to polar coordinates perpendicular to the symmetry axis.Carot (2000) generalized these results and proved some other implicationsof axial symmetry. Remarkably, a group of motions G 2 containing an axialsymmetry must be Abelian. The ‘extended’ regularity conditions (Vanden Bergh and Wils 1985b) do not necessarily hold even when (19.3) issatisfied. A cyclic Killing vector has closed orbits but – contrary to theaxisymmetric case – fixed points and a regular axis are not assumed toexist, cp. Szabados (1987). The possible Bianchi types of a G 3 containinga cyclic symmetry are determined in Barnes (2000).In this chapter we demand that there are two commuting Killing vectorsξ and η,ξ a ;bη b − η a ;bξ b =0, ξ a ξ a < 0, η a η a > 0, (19.4)i.e. they generate an Abelian group G 2 I, see §17.1. Carter (1970) hasshown that the cases of the greatest physical importance, asymptoticallyflat stationary axisymmetric gravitational fields, necessarily admitan Abelian group G 2 , so that for them (19.4) does not impose an additionalrestriction. If the cyclic Killing vector η has a non-null orbit, thanany G 2 containing η is Abelian (Ernotte 1980).The Killing vectors ξ and η are uniquely determined if we demand that(i) η has compact trajectories (in general other subgroups G 1 will havenon-compact trajectories), (ii) η is normalized by (19.3) and (iii) spacetimeis asymptotically flat and ξ is normalized so that F = ξ a ξ a →−1holds in the asymptotic region.19.2 Orthogonal surfacesThe Killing trajectories form two-dimensional orbits T 2 ; the simplebivectorv ab =2ξ [a η b] , v ab v ab < 0, (19.5)