3-manifolds with constant Ricci eigenvalues \((\lambda , \lambda , 0)\)

We consider complete Riemannian 3-manifolds whose Ricci tensors have constant eigenvalues \((\lambda , \lambda , 0)\) $(\lambda ,\lambda ,0)$ . When \(\pi _1\) ${\pi }_1$ is finitely generated, we classify the topology of such manifolds by showing that they have a free fundamental group if non-trivial and that every free group is obtained. We give a description up to isometry when the metric is locally irreducible or when it is analytic.