  If $\mathcal{L}$ is a laminations with hyperbolic singularities, embedded in
a compact homogeneous K\"ahler surface, without directed closed positive
currents. Then, $\mathcal{L}$ has a unique directed positive harmonic current
of mass one. This result was previously obtained by Fornaess and Sibony for
$\mathbb{CP}^2$ and we generalize it for the rest of compact homogeneous
K\"ahler surfaces.
