Integral points on symmetric varieties and Satake compatifications

Let $V$ be an affine symmetric variety defined over $\Bbb Q$. We compute the asymptotic distribution of the angular components of the integral points in $V$. This distribution is described by a family of invariant measures concentrated on the Satake boundary of $V$. In the course of the proof, we describe the structure of the Satake compactifications for general affine symmetric varieties and compute the asymptotic of the volumes of norm balls.