  We study the problem of string propagation in a general instanton background
for the case of the complete heterotic superstring. We define the concept of
generalized HyperK\" ahler manifolds and we relate it to (4,4) superconformal
theories. We propose a generalized h-map construction that predicts a universal
SU(6) symmetry for the modes of the string excitations moving in an instanton
background. We also discuss the role of abstract $N$=4 moduli and, applying it
to the particular limit case of the solvable SU(2) X R instanton found by
Callan et al. we show that it admits deformations and corresponds to a point in
a 16-dimensional moduli space. The geometrical characterization of the other
spaces in the same moduli-space remains an outstanding problem.
