A surface immersed in R4 is called a proper affine sphere if the position vector belongs to the affine normal plane. We classify proper affine spheres with ∇⊥g⊥=0 whose affine mean curvature vector has constant length. Moreover, we find some concrete examples of affine spheres which are not affine umbilical.
 Dillen, F., Mys, G., Verstraelen, L., Vrancken, L.1994The affine mean curvature vector for surfaces in R4Math. Nachr.166155–165.