  Apollonian circle packings arise by repeatedly filling the interstices
between mutually tangent circles with further tangent circles. In Euclidean
space it is possible for every circle in such a packing to have integer radius
of curvature, and we call such a packing an integral Apollonian circle packing.
There are infinitely many different integral packings; these were studied in
the paper \cite{GLMWY21}. Integral circle packings also exist in spherical and
hyperbolic space, provided a suitable definition of curvature is used (see
\cite{LMW02}) and again there are an infinite number of different integral
packings. This paper studies number-theoretic properties of such packings. This
amounts to studying the orbits of a particular subgroup $\sA$ of the group of
integral automorphs of the indefinite quaternary quadratic form $Q_{\sD}(w, x,
y, z)= 2(w^2+x^2 +y^2 + z^2) - (w+x+y+z)^2$. This subgroup, called the
Apollonian group, acts on integer solutions $Q_{\sD}(w, x, y, z)=k$. This paper
gives a reduction theory for orbits of $\sA$ acting on integer solutions to
$Q_{\sD}(w, x, y, z)=k$ valid for all integer $k$. It also classifies orbits
for all $k \equiv 0 \pmod{4}$ in terms of an extra parameter $n$ and an
auxiliary class group (depending on $n$ and $k$), and studies congruence
conditions on integers in a given orbit.
