  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].
