  We give a gauge invariant characterisation of the elliptic affine sphere
equation and the closely related Tzitz\'eica equation as reductions of real
forms of $SL(3, \C)$ anti--self--dual Yang--Mills equations by two
translations, or equivalently as a special case of the Hitchin equation.
  We use the Loftin--Yau--Zaslow construction to give an explicit expression
for a six--real dimensional semi--flat Calabi--Yau metric in terms of a
solution to the affine-sphere equation and show how a subclass of such metrics
arises from 3rd Painlev\'e transcendents.
