 Planar Einstein metric with cubic radial function

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 $z\ne 0$. 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 $12{\alpha }^2{z}^6$, so nonzero $\alpha$ does not give constant sectional curvature despite the Einstein equation.