Christoffel symbol with respect to the target space metric $G_{AB}$.

Weyl-Papapetrou form {#subsec:II-B}
--------------------

Now we consider another Killing vector field $\xi_3$ which is assumed to commute with the other Killing vectors $\xi_a$ and will be identified below as the asymptotic time-translation Killing vector field. Let us consider the condition that the two-dimensional distribution orthogonal to three Killing vector fields $\xi_I\ (I=1,2,3)$ becomes integrable. The commutativity of Killing vector fields, $[\xi_I, \xi_J]=0$, enables us to find coordinate system $x_I$ $(I=1, 2, 3)$, so that $\xi_I=\partial/\partial {x^I}$ and the coordinate components of the metric become independent of $x^I$. We now recall the following theorem about the integrability of two-planes orthogonal to Killing vector fields [@weyl; @Harmark]:

[**Proposition.**]{}

*If three mutually commuting Killing vector fields $\xi_I\ (I=1,2,3)$ in a five-dimensional spacetime satisfy the following two conditions*

1.  $\xi_{1}^{[\mu _1}\xi_{2}^{\mu _2}
    \xi_{3}^{\mu _{2}}D^\nu \xi_{I}^{\rho ]}=0$ holds at at least one point of the spacetime for a given $I=1,2,3$,

2.  $\xi_I^\nu R_\nu ^{[\rho}
    \xi_{1}^{\mu _1}\xi_{2}^{\mu _2}
    \xi_{3}^{\mu _{2}]}=0 $ holds for all $I=1,2,3$,

then the two-planes orthogonal to the Killing vector fields $\xi_I\ (I=1,2,3)$ are integrable.

Note here that one can replace a pair of Killing vector fields $(\xi_1,\xi_2)$ above by another pair $(\xi_2,\xi_3)$. 