Hermitian, Ricci-flat toric metrics on non-compact surfaces \`a la Biquard-Gauduchon

Biquard-Gauduchon have shown that conformally K\"ahler, Ricci-flat, ALF toric metrics on the complement of toric divisors are: the Taub-NUT metric with reversed orientation, in the Kerr-Taub-bolt family or in the Chen-Teo family. The same authors have also given a unified construction for the above families relying on an axi-symmetric harmonic function on $\mathbb{R}^3$. In this work, we reverse this construction and use methods from a paper of the second named author, "Uniqueness among scalar-flat K\"ahler metrics on non-compact toric 4-manifolds", to show that all conformally K\"ahler, Ricci-flat, toric metrics on the complement of toric divisors, under some mild assumptions on the associated moment polytope, are among the families above. In particular all such metrics are ALF.