  Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect
helical symmetry by reducing the Poisson bracket for 3D dynamics to a helically
symmetric one, as an extension of the previous study for translationally
symmetric XMHD (D.A. Kaltsas et al, Phys. Plasmas 24, 092504 (2017)). Four
families of Casimir invariants are obtained directly from the symmetric Poisson
bracket and they are used to construct Energy-Casimir variational principles
for deriving generalized XMHD equilibrium equations with arbitrary macroscopic
flows. The system is then cast into the form of Grad-Shafranov-Bernoulli
equilibrium equations. The axisymmetric and the translationally symmetric
formulations can be retrieved as geometric reductions of the helical symmetric
one. As special cases, the derivation of the corresponding equilibrium
equations for incompressible plasmas is discussed and the helically symmetric
equilibrium equations for the Hall MHD system are obtained upon neglecting
electron inertia. An example of an incompressible double-Beltrami equilibrium
is presented in connection with a magnetic configuration having non-planar
helical magnetic axis.
