Walker manifolds always carry a G-Lie foliation from their null parallel distribution, with Ric vanishing on the distribution and explicit classifications in dimensions 3 and 4.
Molino,Riemannian Foliations, Progress in Mathematics, vol
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On the Lie Foliation structure of Walker Manifolds
Walker manifolds always carry a G-Lie foliation from their null parallel distribution, with Ric vanishing on the distribution and explicit classifications in dimensions 3 and 4.