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- (, ) -Sasaki manifolds and (, ) -Sasaki manifolds, the latter being a convenient reformulation of spin -Einstein -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- (, ) -Sasaki manifolds.