Hamiltonian stationary Lagrangian surfaces in complex space forms with constant-length gradient of the mean curvature

We study Hamiltonian stationary Lagrangian surfaces in complex space forms with nowhere-zero mean curvature vector $H$. We prove that if the Gaussian curvature is constant and $\text{grad}\mathrm{\mid }H\mathrm{\mid }$ has positive constant length, then both the surface and the ambient space are flat. Moreover, the immersion is locally congruent to the known surface obtained as the product of a Cornu spiral and its reflection.