  This paper deals with the $\Gamma$-lattice points problem associated to a
discrete subgroup of motions $\Gamma$ in the complex hyperbolic space
$\mathbb{C} H^n$. We give two integral formulas for the local average of the
number $N(T, z, z')$ of $\Gamma$- lattice points in a sphere of radius $T$ in
$\mathbb{C} H^n$. The first on is in terms of the solution of the
$\Gamma$-automorphic wave equation on $\mathbb{C} H^n$ and the second is given
in terms of the spectral function of the Laplace-Beltrami operator under
$\Gamma$-automorphic boundary conditions. We use the obtained integral formulas
to obtain an asymptotic behavior of the number $N(T, z, z')$ as $T\rightarrow
\infty$, with an estimate of the remainder term. Our principal tools are the
explicit solution of the wave equation on the complex hyperbolic space, special
functions and spectral theory of the Laplace-Beltrami operator under $\Gamma$
automorphic boundary conditions.
