  The Hamiltonian structure of ideal translationally symmetric extended MHD
(XMHD) is obtained by employing a method of Hamiltonian reduction on the
three-dimensional noncanonical Poisson bracket of XMHD. The existence of the
continuous spatial translation symmetry allows the introduction of Clebsch-like
forms for the magnetic and velocity fields. Upon employing the chain rule for
functional derivatives, the 3D Poisson bracket is reduced to its symmetric
counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir
invariants are identified, and used to obtain energy-Casimir variational
principles for generalized XMHD equilibrium equations with arbitrary
macroscopic flows. The obtained set of generalized equations is cast into
Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static
plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria,
where the electron inertia is neglected. The barotropic Hall MHD equilibrium
equations are derived as a limiting case of the XMHD GSB system, and a
numerically computed equilibrium configuration is presented that shows the
separation of ion-flow from electromagnetic surfaces.
