Metrics with prescribed Ricci curvature on homogeneous spaces

Let GG be a compact connected Lie group and HH a closed subgroup of GG. Suppose the homogeneous space G/HG/H is effective and has dimension 3 or higher. Consider a GG-invariant, symmetric, positive-semidefinite, nonzero (0, 2)-tensor field TT on G/HG/H. Assume that HH is a maximal connected Lie subgroup of GG. We prove the existence of a GG-invariant Riemannian metric gg and a positive number cc such that the Ricci curvature of gg coincides with cTcT on G/HG/H. Afterwards, we examine what happens when the maximality hypothesis fails to hold.