The heterotic G₂ system with reducible characteristic holonomy

We construct solutions to the heterotic G₂ system on almost contact metric manifolds with reduced characteristic holonomy. We focus on $3$-(α,δ)-Sasaki manifolds and (α,δ)-Sasaki manifolds, the latter being a convenient reformulation of spin η-Einstein α-Sasaki manifolds. Investigating a $1$-parameter family of G₂-connections on the tangent bundle, we obtain several approximate solutions as well as one new class of exact solutions on degenerate $3$-(α,δ)-Sasaki manifolds.