We have developed a new infinite-order perturbation approach for treating strongly anisotropic magnets. This formalism is based on a canonical transformation of a given system into a new system with an effective two-ion anisotropy which, for example, can be treated by conventional spin-wave techniques. An expression for the spin-wave energy for the general conical magnetic moment configuration with arbitrary single-ion anisotropy has been derived using this formalism. It was found that an analysis of the transverse spin-wave spectra can at most result in a determination of a renormalized exchange integral and two renormalized anisotropy constants. A comparison of the theoretical predictions of this formalism with experimentally determined spin-wave spectra indicate that a substantial part of the large two-ion anisotropy, which had been previously introduced to describe the spin-wave spectra in the heavy rare earth series, actually resulted from the fitting procedure which was used and not from any physical source.