We investigate the spectrum of small amplitude oscillations of the sine-Gordon vortex-antivortex pair in layered ferromagnetic systems described by the anisotropic three-dimensional XY-model. In the case of a small interlayer coupling constant, vortex-pairs can be formed independently in each plane. Analytical expressions for two discrete modes with zero frequency are obtained. These zero modes are associated with the two-dimensional motion of the excitation on an individual plane and are important because the interference between them and a vortex-pair gives rise to contributions to the central peak. The continuum states, which contribute to EPR linewidth, are derived using the Born approximation.