  We present some properties of hyperkahler torsion (or heterotic) geometry in
four dimensions that make it even more tractable than its hyperkahler
counterpart. We show that in $d=4$ hypercomplex structures and weak torsion
hyperkahler geometries are the same. We present two equivalent formalisms
describing such spaces, they are stated in the propositions of section 1. The
first is reduced to solve a non-linear system for a doublet of potential
functions, first found by Plebanski and Finley. The second is equivalent to
finding the solutions of a quadratic Ashtekar-Jacobson-Smolin like system, but
without a volume preserving condition. This is why heterotic spaces are simpler
than usual hyperkahler ones. We also analyze the strong version of this
geometry. Certain examples are presented, some of them are metrics of the
Callan-Harvey-Strominger type and others are not. In the conclusion we discuss
the benefits and disadvantages of both formulations in detail.
