results of the previous section are general enough to encompass the special case of type D
backgrounds, the tetrad choice we made (with -r = 0) is incompatible with the complete
integration of the background field equations which is possible in type D spacetimes [45].
The complete integration requires that each of 1" and n" be aligned with one of the two
PNDs. In that case we can exploit the full power of the Goldbergf-Sachs theorem and its
corollaries to set a = K' = a = a' = ~o = ~1 = ~3 = 4 = 0, while maintaining -r / 0
and -r' / 0. In this section we repeat the previous calculation with this different choice of
tetrad.
The result of integrating SiI = 0 is the same as in the case of a type II background,
given in Equation 4-16. The residual gauge vector, however, now has the following, more
complex, form (details of the integration are given in Appendix C):
Cl = Clo
1 1 1 P1 1\
2 2 2 p2 2
[xo go1 1 1 ~
p p 2 p p
-[7Op(8 UO + so .48' + o (o em p reo (4-25)
o 1 1 o 1 1
p2 P2
1l1
1m=~ ;~ 1 c" o
m mo_ o l o P1
p p n~ i~~ I7:0 il(y&