  In this paper, we fix the complex structure and explore the moduli space of
the heterotic system by considering two different yet "dual" deformation paths
starting from a K\"ahler solution. They correspond to deformation along the
Bott-Chern cohomology class and the Aeppli cohomology class respectively. Using
the implicit function theorem, we prove the local existence of heterotic
solutions along these two paths and hence establish an initial step to
construct local moduli coordinates around a K\"ahler solution.
