Geodesic ball packings generated by rotations and monotonicity behavior of their densities in $\mathbf{H}^2\!\times\!\mathbf{R}$ space

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.