Let $ (M,g) $ be a Riemannian manifold. A gradient conformal vector field on $ M $ is a conformal vector field $ X $ which is at the same time the gradient of a function on $ M $ : \begin{equation} X=\nabla f , L_{X}g=2\rho g \end{equation} where $L$ is the Lie derivative. Is there always gradient conformal vector field on $M$? Thanks.
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