The heterotic G${}_2$ system with reducible characteristic holonomy

We construct solutions to the heterotic G${}_2$ system on almost contact metric manifolds with reduced characteristic holonomy. We focus on $3$-$(\alpha ,\delta )$-Sasaki manifolds and $(\alpha ,\delta )$-Sasaki manifolds, the latter being a convenient reformulation of spin $\eta$-Einstein $\alpha$-Sasaki manifolds. Investigating a $1$-parameter family of G${}_2$-connections on the tangent bundle, we obtain several approximate solutions as well as one new class of exact solutions on degenerate $3$-$(\alpha ,\delta )$-Sasaki manifolds.