  We construct smooth families of compact special Lagrangian submanifolds
embedded in some toric hyper-K\"ahler manifolds, which never become
holomorphic Lagrangian submanifolds via any hyper-K\"ahler rotations. These
families converge to special Lagrangian immersions with self-intersection
points in the sense of current. To construct them, we apply the
desingularization method developed by Joyce.
