Does the Poynting vector always represent electromagnetic power flow?

Abstract: Poynting vector as electromagnetic power flow has prevailed over one hundred years in the community. However in this paper, it is shown from Maxwell equations that the Poynting vector may not represent the electromagnetic power flow for a plane wave in a non-dispersive, lossless, non-conducting, anisotropic uniform medium; this important conclusion revises the conventional understanding of Poynting vector. It is also shown that this conclusion is clearly supported by Fermat's principle and special theory of relativity.