Examples of almost Einstein structures on products and in cohomogeneity one

Almost Einstein manifolds are conformally Einstein up to a scale singularity, in general. This notion comes from conformal tractor calculus. In the current paper we discuss almost Einstein structures on closed Riemannian product manifolds and on 4-manifolds of cohomogeneity one. Explicit solutions are found by solving ordinary differential equations. In particular, we construct three families of closed 4-manifolds with almost Einstein structure corresponding to the boundary data of certain unimodular Lie groups. Two of these families are Bach-flat, but neither (globally) conformally Einstein nor half conformally flat. On products with a 2-sphere we find an exotic family of almost Einstein structures with hypersurface singularity as well.