  Let G be the identity component of SO(n,1), acting linearly on a finite
dimensional real vector space V. Consider a vector w_0 in V such that the
stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line
Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup
Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the
number of points in w_0Gamma of norm at most T, provided that the
Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the
Gamma skinning size of w_0 are finite.
  The main ergodic ingredient in our approach is the description for the
limiting distribution of the orthogonal translates of a totally geodesically
immersed closed submanifold of Gamma\H^n. We also give a criterion on the
finiteness of the Gamma skinning size of w_0 for Gamma geometrically finite.
