  In the present work, we investigate classical solutions of the
Maxwell-Carroll-Field-Jackiw-Proca (MCFJP) electrodynamics for the cases a
purely timelike and spacelike Lorentz-violating (LV) background. Starting from
the MCFJP Lagrangian and the associated wave equations written for the
potential four-vector, the tensor form of the Green function is achieved. In
the timelike case, the components of the stationary Green function are
explicitly written. The classical solutions for the electric and magnetic field
strengths are then evaluated, being observed that the electric sector is not
modified by the LV background, keeping the Maxwell-Proca behavior. The magnetic
field associated with a charge in uniform motion presents an oscillating
behavior that also provides an oscillating MCFJ solution (in the limit of a
vanishing Proca mass), but does not recover the Maxwell-Proca solution in the
limit of vanishing background. In the spacelike case, the stationary Green
function is written and also explicitly carried out in the regime of a small
background. The electric and magnetic fields reveal to possess an exponentially
decaying behavior, that recover the Maxwell-Proca solutions.
