Constant mean curvature hypersurfaces in ${\mathbb{H}}^2\times {\mathbb{H}}^2$ with double horocyclic symmetry

We study constant mean curvature hypersurfaces in ${\mathbb{H}}^2\times {\mathbb{H}}^2$ invariant under a double horocyclic action. We show that the CMC condition reduces to a single autonomous ordinary differential equation for an angular function. From this reduction, we obtain three distinct regimes and solve the ODE explicitly in each case, obtaining an existence and uniqueness result for double horocyclic CMC hypersurfaces in ${\mathbb{H}}^2\times {\mathbb{H}}^2$. Finally, we classify the equilibrium solutions and identify the corresponding homogeneous models: ${\mathbb{H}}^3$, ${\mathbb{H}}^2\times \mathbb{R}$, ${\mathrm{S}\mathrm{o}\mathrm{l}}_3$, and left-invariant metrics on semidirect product Lie groups.