Abstract: Proper Definite Affine Spheres (or PDAS-surfaces) in R^3 are equivalent to primitive harmonic maps from Riemann surfaces into the 6-symmetric space SL(3,R)/SO(2,R). After proving an Iwasawa-type double coset decomposition for the associated complex twisted loop group of A^(2)_2 type, we give a Weierstrass-type representation by either holomorphic or other meromorphic DPW potentials. The linear DPW flow on the only two open Iwasawa cosets corresponds exactly to the elliptic and the hyperbolic parts respectively, while the other cosets having finite codimensions provide algebraic types for the branch curves or points between these parts. Then equivariant PDAS-surfaces are classified: among all seven Riemann surfaces admitting one-parameter group of automorphisms, the Riemann sphere, the complex plane, the punctured plane, the unit disk and the annulus admit affine conformal branch immersions into R^3 as global equivariant PDAS-surfaces; but the punctured disk and any torus do not admit such immersions. There is no equivariant PDAS-surface with screw-motion symmetry either. Most of these global surfaces consist of both elliptic and hyperbolic parts. They suggest that the Calabi conjecture for complete hyperbolic affine spheres (proved by Cheng-Yau) may be generalized to some global PDAS-surfaces of mixed parts. This is a joint work with Josef Dorfmeister and Gang Wang.