From the introduction: We construct a single complex differential equation for a complex three-dimensional vector \(A\)-field which is equivalent to the system of Maxwell equations in the presence of electric and magnetic charges and currents. To analyze this equation referred to as the Hamiltonian form of the Maxwell equations, we use methods of distribution theory. This approach allows one to construct and analyze discontinuous solutions with jumps not only in derivatives but also in the unknown functions themselves. This is important in the investigation of nonstationary wave processes of shock wave type.