A study of the linear dispersion relation for left-handed transverse electromagnetic waves in a solar-wind-like magnetized ion background is presented. In this analysis the dispersion characteristics of the high-order modes of wave propagation are included. These modes exist below but also above the proton gyrofrequency, and its dispersion relation can be approximated by an acoustic-like one. It is shown that a discontinuity arises in the dispersion relation for a proton alpha particles magnetized plasma, when the fraction of alpha particles is increased by virtue of coupling between the normal ion cyclotron mode and high-order modes. It is also shown that the discontinuity can disappear by changing the anisotropy of the alpha particles or their drift velocity. In that case the normal ion cyclotron mode is restored. Some implications of these modes in heating the plasma are also considered. ¿ American Geophysical Union 1996