PROJECTIVELY FLAT AFFINE SURFACES THAT ARE NOT LOCALLY SYMMETRICAL

By studying affine rotation surfaces (ARS), we prove that any surface affine congruent to x(2) + epsilon y(2) = z(r) or y(2) = z(X + epsilon z log z) is projectively flat but is neither locally symmetric nor an affine sphere, where epsilon is 1 or -1, r is an element of R-{-1, 0, 1, 2},and z > 0. The significance of these surfaces is due to the fac t that until now x(2) + epsilon y(2) = z(-1) are the only known surfac es which are projectively flat but not locally symmetric. Although Pod esta recently proved the existence of an affine surface satisfying the above italicized conditions, he did not construct any concrete exampl e.