According to Singer [Si] a Riemannian manifold (M, g) is said to be curvature homogeneous if, for every two points, p, q~M, there is a linear isometry F : T p M ~ T q M between the corresponding tangent spaces such that F* Rq=Rp (where R denotes the curvature tensor of type (0, 4)). Note that a (locally) homogeneous Riemannian manifold is automatically curvature homogeneous. Explicit locally nonhomogeneous examples have been constructed by many authors ([Sel, T, Ya, K-T-V1-K-T-V3, K1-K3]; see especially [K-T-V2] and [K-T-V3] for more complete references). For 3-dimensional Riemannian manifolds (M, 9) the following simple criterion holds: (M, g) is curvature homogeneous if and only if all Ricci eigenvalues of (M, g) are constant. Thus the problem to classify all 3-dimensional Riemannian manifolds with prescribed constant Ricci eigenvalues is of considerable interest. This problem was investigated already in 1916 by Bianchi [B] who made a classification under a strong additional hypothesis of "normality". He found only some homogeneous Riemannian spaces as solutions. On the other hand, the following conclusion follows from an observation by Milnor [M] and a result by Sekigawa [Se2]: For a homogeneous Riemannian 3-manifold (M, g), the signature of the Ricci tensor is never equal to ( +, +, ) or (+,0, -). Let us describe shortly what is known about the problem at the present. The case in which all Ricci eigenvalues are equal is trivial we obtain only spaces of constant curvature. The case pl = P2 4= P3 was solved completely by the first author in [K 1] and [K2]. The answer is that the local isometry classes of the corresponding metrics always depend on two arbitrary functions of one variable. (Notice that the local isometry classes of locally homogeneous spaces