= Linear dilaton counterexample to zero integration constant
In flat target space, $\varphi=q_aX^a$ has zero Hessian and therefore solves the <leading metric-dilaton Weyl condition> for every constant covector $q$. But $\Box\varphi-2|\nabla\varphi|^2=-2q^2$, which need not vanish. This demonstrates the integration constant left by the <contracted Bianchi identity>. Full quantum Weyl invariance fixes the allowed $q^2$ through the <central charge>, as in a <linear dilaton conformal field theory>; the metric equation alone does not.
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