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 curvatire surfaces means the hyperbolic affine spheres. Let M be a surface with constant centro-affine curvatire surface 1, locally given by the graph $(x, f(x))$, then by the Legendre transformation relative to $f$, the equation is $$\det(D^2 u)=(-u)^{-4}.$$
The answer can be found in this book (proved by Minkowski integral formulas, and hold with more larger class).