On Hamiltonian stationary Lagrangian spheres in non-Einstein Kaehler surfaces

Hamiltonian stationary Lagrangian spheres in Kaehler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kaehler surfaces given by the product ${\mathrm{\Sigma }}_1\times {\mathrm{\Sigma }}_2$ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example is defined when the surfaces ${\mathrm{\Sigma }}_1$ and ${\mathrm{\Sigma }}_2$ are spheres.