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There are two free coupling parameters $c_{13}$ and $c_{14}$ in theEinstein-\AE ther metric describing a non-rotating black hole. This metric isthe 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 inan external asymptotically uniform magnetic field, the Hamiltonian systemdescribing the motion of charged particles around the black hole is notintegrable. However, the Hamiltonian allows for the construction of explicitsymplectic integrators. The proposed fourth-order explicit symplectic scheme isused to investigate the dynamics of charged particles because it exhibitsexcellent long-term performance in conserving the Hamiltonian. No universalrule can be given to the dependence of regular and chaotic dynamics on varyingone or two parameters $c_{13}$ and $c_{14}$ in the two cases of $0\leq2c_{13}<c_{14}<2$ and $0\leq c_{14}<2c_{13}<2$. The distributions of order andchaos in the binary parameter space $(c_{13},c_{14})$ rely on differentcombinations of the other parameters and the initial conditions.

Effects of coupling constants on chaos of charged particles in the $Einstein-\AE$ ther theory