internal spaces $X^6=\ $Sp(2)/Sp(1)$\times$U(1) or SU(3)/U(1)$\times$U(1) were considered (see also [@Toma]) and it was found that the Kaluza-Klein decoupling for the original AdS$_4$ vacua requires that the above-mentioned internal spaces are substituted by the nearly Calabi-Yau spaces Sp(1,1)/Sp(1)$\times$U(1), SU(1,2)/U(1)$\times$U(1) or by their compact analogues obtained by quotienting out the internal manifolds by a discrete group. In our paper, we describe various solutions of the Hermitian-Yang-Mills equations on all these four coset spaces $X^6$ that can further be used in heterotic string compactifications with non-trivial background fluxes [@FL]-[@Zoupanos].

Since their discovery, more than ten years ago, tractable flux compactifications in string theory have become a very active area of research (see e.g. [@Grana; @DoKa; @BKLS] for reviews and references). This has been particularly explored in type IIB theory (see e.g. [@GKP]-[@LRS]), and some efforts have been devoted to moduli-fixing problem in the case of type IIA compactifications (see e.g. [@LoMi; @DKPZ; @GL]) where metric fluxes can arise partially from the T-duality of NS fluxes [@GLMW; @KSTT]. Compactifications in the presence of fluxes can be described in the language of $G$-structures on $d$-dimensional manifolds: SU(3) structure for dimension $d{=}6$, $G_2$ for $d{=}7$ and Spin(7) for $d{=}8$ (see e.g. [@GMPW]-[@XuPhD] and references therein). 