  This article carries out a qualitative analysis on a system of integral
equations of the Hardy--Sobolev type. Namely, results concerning Liouville type
properties and the fast and slow decay rates of positive solutions for the
system are established. For a bounded and decaying positive solution, it is
shown that it either decays with the slow rates or the fast rates depending on
its integrability. Particularly, a criterion for distinguishing integrable
solutions from other bounded and decaying solutions in terms of their
asymptotic behavior is provided. Moreover, related results on the optimal
integrability, boundedness, radial symmetry and monotonicity of positive
integrable solutions are also established. As a result of the equivalence
between the integral system and a system of poly-harmonic equations under
appropriate conditions, the results translate over to the corresponding
poly-harmonic system. Hence, several classical results on semilinear elliptic
systems are recovered and further generalized.
