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

A stability of vector bundles with twisted sections and the Donaldson-Thomas instantons on compact K\"{a}hler threefolds