You can see the book by Li-Simon-Zhao:"Global Affine differetial geometry of hypersurfaces. Berlin: Walter de Gruyter, 1993." I think yours definition of the constant centro-affine curvature surfaces means the ellptic affine spheres. Let M be a surface with constant centro-affine curvature 1, locally given by the graph $(x, f(x))$, then by the Legendre transformation relative to $f$, the equation is (with center at the orgin) $$\det(D^2 u)=(u)^{-4}.$$ The answer can be found in this book, due to Blaschke, Deicke, Calabi.
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