  In this paper, using the Milnor-type theorem technique, we provide on each
nilpotent five dimensional Lie group, some global existence result 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 and T
a given left-invariant symmetric (0, 2)-tensor field.
