  In this paper, we study the distribution of integral points on parametric
families of affine homogeneous varieties. By the work of Borel and
Harish-Chandra, the set of integral points on each such variety consists of
finitely many orbits of arithmetic groups, and we establish an asymptotic
formula (on average) for the number of the orbits indexed by their Siegel
weights. In particular, we deduce asymptotic formulas for the number of
inequivalent integral representations by decomposable forms and by norm forms
in division algebras, and for the weighted number of equivalence classes of
integral points on sections of quadratic surfaces. Our arguments use the
exponential mixing property of diagonal flows on homogeneous spaces.
