Invariant scalar-flat K\"ahler metrics on line bundles over generalized flag varieties

Let $G$ be a simply-connected semisimple compact Lie group, $X$ a compact K\"ahler manifold homogeneous under $G$, and $L$ a negative $G$-equivariant holomorphic line bundle over $X$. We prove that all $G$-invariant K\"ahler metrics on the total space of $L$ arise from the Calabi ansatz. Using this, we then show that there exists a unique $G$-invariant scalar-flat K\"ahler metric in each K\"ahler class of $L$.