  We introduce and study the equiaffine symmetric {\bf hyperspheres}. For the
first step we consider the locally strongly convex ones. In fact, by the idea
used by Naitoh, we provide in this paper a direct proof of the complete
classification for those affine symmetric hyperspheres. Then, via an earlier
result of the first author, we are able to provide an alternative proof for the
classification theorem of the affine hypersurface with parallel Fubini-Pick
forms, which has already been established by Z.J. Hu et al in a totally
different way.
