Einstein-Kahler metrics on symmetric toric Fano manifolds

Let X be a complex toric Fano n-fold and N(T) the normalizer of a maximal torus T in the group of biholomorphic automorphisms Aut(X). We call X symmetric if the trivial character is a single N(T)-invariant algebraic character of T. Using an invariant a,(X) introduced by Tian, we show that all symmetric toric Fano n-folds admit an Einstein-Kahler metric. We remark that so far one doesn't know any example of a toric Fano n-fold X such that Aut(X) is reductive, the Futaki character of X vanishes, but X is not symmetric.