  Let G be a three-dimensional unimodular Lie group, and let T be a
left-invariant symmetric (0, 2)-tensor field on G. We provide the necessary and
sufficient conditions on T for the existence of a pair (g, c) consisting of a
left-invariant Riemannian metric g and a positive constant c such that Ric(g) =
cT, where Ric(g) is the Ricci curvature of g. We also discuss the uniqueness of
such pairs and show that, in almost all cases, there exists at most one
positive constant c such that Ric(g) = cT is solvable for some left-invariant
Riemannian metric g.
