Global conformal gauge

Question

I'm reading the Blumenhagen-Lust-Theisen book on string theory. On page 18, They want to discuss whether a global conformal flat metric can exist, namely $$ h_{\alpha\beta}=e^{2\phi}\eta_{\alpha\beta} $$ To see if this is possible, they decompose the change of metric under reparametrizations and Weyl rescaling as $$ \delta h_{\alpha\beta}=-(P\xi)_{\alpha\beta}+2\tilde\Lambda h_{\alpha\beta} $$ where $$ (P\xi)_{\alpha\beta}:=\nabla_\alpha\xi_\beta+\nabla_\beta\xi_\alpha-(\nabla_\gamma\xi^\gamma)h_{\alpha\beta} $$ and $\xi$ is given by the reparametrization $X\to X+\xi$. The authors say that the first term in $\delta h_{\alpha\beta}$ is traceless, and the second term has a non-zero trace, so there must exist a globally defined vector field $\xi^\alpha$ such that $$ (P\xi)_{\alpha\beta}=t_{\alpha\beta} $$ for all symmetric traceless $t_{\alpha\beta}$. But I don't understand how this implication holds, and how does the existence of such vector field related to the existence of global conformal metric.