Capillary Immersions in E($\kappa$,$\tau$)

In [3] and [11] the authors showed the existence of a Codazzi pair defined on any constant mean curvature surface in the homogeneous spaces E($\kappa$,$\tau$) associated to the Abresch-Rosenberg differential. In this paper, we use the mentioned Codazzi pair to classify capillary disks in E($\kappa$,$\tau$). As a consequence, the results presented in this paper generalize the previous classification of constant mean curvature disks in the product spaces ${\mathbb{S}}^2\times \mathbb{R}$ and $$\mathbb{H}$^2 \times $\mathbb{R}$$ in [4] and [5].