  It is shown that a canonical geometric setting of the integrable TED equation
is a Kahlerian tangent bundle of an affine manifold. The remarkable
multi-dimensional consistency of this 4+4-dimensional dispersionless partial
differential equation arises naturally in this context. In a particular
4-dimensional reduction, the affine manifolds turn out to be self-dual Einstein
spaces of neutral signature governed by Plebanski's first heavenly equation. In
another reduction, the affine manifolds are Hessian, governed by compatible
general heavenly equations. The Legendre invariance of the latter gives rise to
a (dual) Hessian structure. Foliations of affine manifolds in terms of
self-dual Einstein spaces are also shown to arise in connection with a natural
5-dimensional reduction.
