  The problem of characterizing conformally Einstein manifolds by tensorial
conditions has been tackled recently in papers by M. Listing, and in work by A.
R. Gover and P. Nurowski. Their results apply to metrics satisfying a
"non-degeneracy" condition on the Weyl tensor \W. We investigate the geometry
of the foliations arising on conformally Einstein spaces (with Riemannian
signature) where this condition fails, which then allows us to characterize a
general class of locally conformally Einstein Riemannian manifolds with
degenerate Weyl tensor.
