Quadratic field equations on the brane

It is shown that four dimensional vacuum Einstein solutions simply embedded in five dimensions obey the Gauss-Bonnet-Einstein field equations: $G_{ab}+\alpha G{B}_{ab}+{\delta }_{ab}^{55}\alpha \exp (-2\chi \mathrm{/}\sqrt{\alpha })G{B}_4=0$ and the Pauli-Einstein equations $G_{ab}-3\alpha P_{ab}\mathrm{/}5=0$, 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.