Noncompact Iwasawa factorization and translationally equivariant hyperbolic affine spheres

We establish the noncompact Iwasawa factorization for a Delaunay-type potential associated with affine spheres on the complex plane away from countably many lines. Using the DPW method, we give an explicit description of the factorization in terms of Weierstrass elliptic functions via a reduction to a linear system related to the Tzitz\'eica equation. As an application, we construct explicit translationally equivariant hyperbolic affine spheres and classify them according to their slice curves. In particular, we show that every such affine sphere is equiaffinely equivalent to one whose slice curve is a circle, hyperbola, or parabola, consistent with the Calabi correspondence between hyperbolic affine spheres and proper convex cones.