  A small deformation controlled by four free parameters to the Schwarzschild
metric could be referred to a nonspinning black hole solution in alternative
theories of gravity. Because such a non-Schwarzschild metric can be changed
into a Kerr-like black hole metric via a complex coordinate transformation, the
recently proposed time-transformed explicit symplectic integrators for the Kerr
type spacetimes are suitable for a Hamiltonian system describing the motion of
charged particles around the non-Schwarzschild black hole surrounded with an
external magnetic field. The obtained explicit symplectic methods are based on
a time-transformed Hamiltonian split into seven parts, whose analytical
solutions are explicit functions of new coordinate time. Numerical tests show
that such explicit symplectic integrators for intermediate time-steps perform
good long-term performance in stabilizing Hamiltonian errors regardless of
regular or chaotic orbits. One of the explicit symplectic integrators with the
techniques of Poincar\'{e} sections and fast Lyapunov indicators is applied to
investigate the effects of the parameters including the four free deformation
parameters on the orbital dynamical behavior. From the global phase-space
structure, chaotic properties are typically strengthened under some
circumstances as any one of the energy and the magnitudes of magnetic parameter
and negative deformation parameters increases. However, they are weakened when
each of the angular momentum and positive deformation parameters increases.
