  We consider the heterotic string on Calabi-Yau manifolds admitting a
Strominger-Yau-Zaslow fibration. Upon reducing the system in the
$T^3$-directions, the Hermitian Yang-Mills conditions can then be reinterpreted
as a complex flat connection on $\mathbb{R}^3$ satisfying a certain co-closure
condition. We give a number of abelian and non-abelian examples, and also
compute the back-reaction on the geometry through the non-trivial
$\alpha'$-corrected heterotic Bianchi identity, which includes an important
correction to the equations for the complex flat connection. These are all new
local solutions to the Hull-Strominger system on $T^3\times\mathbb{R}^3$. We
also propose a method for computing the spectrum of certain non-abelian models,
in close analogy with the Morse-Witten complex of the abelian models.
