= Planar Einstein metric with cubic radial function
{title2=$ds^2=z^{-2}[-Fdt^2+dx^2+dy^2+F^{-1}dz^2],\quad F=1-\alpha z^3$}
This metric has $R_{ab}=-3g_{ab}$ and <cosmological constant> $\Lambda=-3$ in unit curvature-radius normalization. Its regular static coframe is real when $F>0$ and $z\ne0$. Setting $q=\alpha z^3$, its six independent <curvature 2-form> coefficients are $-(1-q)$ in the $03$ and $12$ planes and $-(1+q/2)$ in the other four planes. Their Ricci contractions cancel $q$. The <Weyl tensor> squared invariant is $12\alpha^2z^6$, so nonzero $\alpha$ does not give constant sectional curvature despite the Einstein equation.
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