 Linear dilaton counterexample to zero integration constant

In flat target space, $\phi =q_a{X}^a$ has zero Hessian and therefore solves the leading metric-dilaton Weyl condition for every constant covector $q$. But $\square \phi -2|\nabla \phi {|}^2=-2q^2$, 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.