  We consider a version of Hermitian-Einstein equation but perturbed by a Higgs
field with a solution called a Donaldson-Thomas instanton on compact K\"ahler
threefolds. The equation could be thought of as a generalization of the Hitchin
equation on Riemann surfaces to K\"ahler threefolds. In the appendix of
arXiv:0805.2192, following an analogy with the Hitchin equation, we introduced
a stability condition for a pair consisting of a locally-free sheaf over a
compact K\"ahler threefold and a section of the associated sheaf of the
endomorphisms tensored by the canonical bundle of the threefold. In this
article, we prove a Hitchin--Kobayashi-type correspondence for this and the
Donaldson-Thomas instanton on compact K\"ahler threefolds.
