Consider a Kähler manifold with complex structure $J$. Is there a characterization of real-valued functions $H$ for which the corresponding Hamiltonian vector field $X_H$ is real-holomorphic, that is, $\mathscr{L}_{X_H}J=0$ ?
Stated differently, is it possible to characterize energy functions on Kähler manifolds that give rise to isometric Hamiltonian dynamics?
