For weak anisotropy there are "out-of-plane vortices" which exhibit a localized structure of the spin components perpendicular to the plane. The vortex equation of motion is 1st order and contains a velocity-dependent "gyrotropic" force (Thiele 1973). We generalize this by allowing changes of the vortex shape due to accelerations and derive a 3rd order equation of motion. It predicts a beat oscillation around the mean vortex path obtained from Thiele's equation. The two frequencies of the beat can be identified with the dominant doublet of the spectrum which we observe in computer simulations. Additional, much weaker doublets can be explained by 5th, 7th, ... order equations.
More generally, we show that the equations of motion for "gyrotropic excitations" form a hierarchy of odd-order equations, while the equations for non-gyrotropic excitations (e.g. kinks or planar vortices) are of even order.