  Given a symmetric variety Y defined over the rationals and a non-zero
polynomial with integer coefficients, we use techniques from homogeneous
dynamics to establish conditions under which the polynomial can be made r-free
for a Zariski dense set of integral points on Y. We also establish an
asymptotic counting formula for this set. In the special case that Y is a
quadric hypersurface, we give explicit bounds on the size of r by combining the
argument with a uniform upper bound for the density of integral points on
general affine quadrics.
