  In \cite{Sz13-1} we defined and described the {\it regular infinite or
bounded} $p$-gonal prism tilings in $\SLR$ space. We proved that there exist
infinitely many regular infinite $p$-gonal face-to-face prism tilings
$\cT^i_p(q)$ and infinitely many regular bounded $p$-gonal non-face-to-face
prism tilings $\cT_p(q)$ for integer parameters $p,q;~3 \le p$, $
\frac{2p}{p-2} < q$. Moreover, in \cite{MSz14} and \cite{MSzV13} we have
determined the symmetry group of $\cT_p(q)$ via its index 2 rotational
subgroup, denoted by $\mathbf{pq2_1}$ and investigated the corresponding
geodesic and translation ball packings.
  In this paper we study the structure of the regular infinite or bounded
$p$-gonal prism tilings, prove that the side curves of their base figurs are
arcs of Euclidean circles for each parameter. Moreover, we examine the
non-periodic geodesic ball packings of congruent regular non-periodic prism
tilings derived from the regular infinite $p$-gonal face-to-face prism tilings
$\cT^i_p(q)$ in $\SLR$ geometry. We develop a procedure to determine the
densities of the above non-periodic optimal geodesic ball packings and apply
this algorithm to them. We look for those parameters $p$ and $q$ above, where
the packing density large enough as possible. Now, we obtain larger density
$\approx 0.626606$ for $(p, q) = (29,3)$ then the maximal density of the
corresponding periodical geodesic ball packings under the groups
$\mathbf{pq2_1}$.
  In our work we will use the projective model of $\SLR$ introduced by E.
{Moln\'ar} in \cite{M97}.
