  It is shown that four dimensional vacuum Einstein solutions simply embedded
in five dimensions obey the Gauss-Bonnet-Einstein field equations:
$G_{ab}+\alpha GB_{ab}+\delta^{55}_{ab}\alpha\exp(-2\chi/\sqrt{\alpha})GB_4=0$
and the Pauli-Einstein equations $G_{ab}-3\alpha P_{ab}/5=0$, and the
Bach-Einstein equations $B_{ab}=0$. General equations are calculated for which
these and similar results follow. It is briefly argued that such field
equations could be significant on large distance scales.
