Uniform K-stability of G-varieties of complexity 1

Let k be an algebraically closed field of characteristic 0 and G a connected, reductive, linear algebraic group of simply connected type over k. Let X be a projective G-variety of complexity 1. We classify G-equivariant normal test configurations of X with integral central fibre via the combinatorial data. We also give a formula of anti-canonical divisors on X. Based on this formula, when X is Q-Fano, we give an expression of the Futaki invariant, and derive a criterion of uniform K-stability in terms of the combinatorial data.