Bismut Ricci flat manifolds with symmetries

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification spaces dimension $5$ also provided. Moreover, we investigate with non-trivial third Betti number, point out other possible ways to manifolds. Finally, since connections correspond fixed points flow, discuss stability some our under flow.