  Suppose that a polarised K\"ahler manifold $(X,L)$ admits an extremal metric
$\omega$. We prove that there exists a sequence of K\"ahler metrics $\{
\omega_k \}_k$, converging to $\omega$ as $k \to \infty$, each of which
satisfies the equation $\bar{\partial} \text{grad}^{1,0}_{\omega_k} \rho_k
(\omega_k)=0$; the $(1,0)$-part of the gradient of the Bergman function is a
holomorphic vector field.
