  There are two free coupling parameters $c_{13}$ and $c_{14}$ in the
Einstein-\AE ther metric describing a non-rotating black hole. This metric is
the Reissner-Nordstr\"{o}m black hole solution when $0\leq 2c_{13}<c_{14}<2$,
but it is not for $0\leq c_{14}<2c_{13}<2$. When the black hole is immersed in
an external asymptotically uniform magnetic field, the Hamiltonian system
describing the motion of charged particles around the black hole is not
integrable. However, the Hamiltonian allows for the construction of explicit
symplectic integrators. The proposed fourth-order explicit symplectic scheme is
used to investigate the dynamics of charged particles because it exhibits
excellent long-term performance in conserving the Hamiltonian. No universal
rule can be given to the dependence of regular and chaotic dynamics on varying
one or two parameters $c_{13}$ and $c_{14}$ in the two cases of $0\leq
2c_{13}<c_{14}<2$ and $0\leq c_{14}<2c_{13}<2$. The distributions of order and
chaos in the binary parameter space $(c_{13},c_{14})$ rely on different
combinations of the other parameters and the initial conditions.

General Relativity and Quantum Cosmology;Astrophysics of Galaxies