  As a generalization of Kahler-Einstein metrics for Fano manifolds with
nonvanishing Futaki invariant, Mabuchi solitons are critical points of a
Calabi-type energy functional. We study their existence on toric Fano varieties
and the underlying algebraic stability notion: relative Ding stability. As a
toy model for a YTD type correspondence, a new feature is the emergence of a
non-uniformly stable case. We show a partial coercivity for the modified Ding
functionals in this case, and obtain singular Mabuchi solitons via a
variational approach. In the unstable case, we determine the maximal
destabilizer which is a simple convex function over the moment polytope, and
establish a Moment-Weight equality which connects the infimum of a Calabi-type
energy and the Berman-Ding invariant.
