  Hamiltonian stationary Lagrangian submanifolds (HSLAG) are a natural
generalization of special Lagrangian manifolds (SLAG). The latter only make
sense on Calabi-Yau manifolds whereas the former are defined for any almost
K\"ahler manifold. Special Lagrangians, and, more specificaly, fibrations by
special Lagrangians play an important role in the context of the geometric
mirror symmetry conjecture. However, these objects are rather scarce in nature.
On the contrary, we show that HSLAG submanifolds, or fibrations, arise quite
often. Many examples of HSLAG fibrations are provided by toric K\"ah-ler
geometry. In this paper, we obtain a large class of examples by deforming the
toric metrics into non toric almost K\"ahler metrics, together with HSLAG
submanifolds.
