  After having investigated several types of geodesic ball packings in
$\mathbf{S}^2 \times \mathbf{R}$ space, in this paper we study the locally
optimal geodesic of simply and multiply transitive ball packings with equal
balls to the space groups generated by rotations in $\mathbf{H}^2 \times
\mathbf{R}$ geometry. These groups can be derived by direct product of the
isometries on hyperbolic plane $\mathbf{H}^2$ and the real line $\mathbf{R}$.
Moreover, we develop a procedure to determine the densities of the above
locally densest geodesic ball packing configurations. Additionally, we examine
the monotonicity properties of the densities within infinite series of the
considered space groups. E. {Moln\'ar} showed, that the homogeneous 3-spaces
have a unified interpretation in the projective 3-sphere
$\mathcal{PS}^3(\mathbf{V}^4,\boldsymbol{V}_4, \mathbf{R})$. In our work, we
use this projective model of $\mathbf{H}^2 \times \mathbf{R}$ to visualize the
locally optimal ball arrangements.
