  We present two generalized hybrid kinetic-Hall magnetohydrodynamics (MHD)
models describing the interaction of a two-fluid bulk plasma, which consists of
thermal ions and electrons, with energetic, suprathermal ion populations
described by Vlasov dynamics. The dynamics of the thermal components are
governed by standard fluid equations in the Hall MHD limit with the electron
momentum equation providing an Ohm's law with Hall and electron pressure terms
involving a gyrotropic electron pressure tensor. The coupling of the bulk,
low-energy plasma with the energetic particle dynamics is accomplished through
the current density (current coupling scheme; CCS) and the ion pressure tensor
appearing in the momentum equation (pressure coupling scheme; PCS) in the first
and the second model, respectively. The CCS is a generalization of two
well-known models, because in the limit of vanishing energetic and thermal ion
densities we recover the standard Hall MHD and the hybrid
kinetic-ions/fluid-electron model, respectively. This provides us with the
capability to study in a continuous manner the global impact of the energetic
particles in a regime extending from vanishing to dominant energetic particle
densities. The noncanonical Hamiltonian structures of the CCS and PCS, which
can be exploited to study equilibrium and stability properties through the
energy-Casimir variational principle, are identified. As a first application
here, we derive a generalized Hall MHD Grad--Shafranov--Bernoulli system for
translationally symmetric equilibria with anisotropic electron pressure and
kinetic effects owing to the presence of energetic particles using the PCS.
