  This paper examines systems of poly-harmonic equations of the Hardy--Sobolev
type and the closely related weighted systems of integral equations involving
Riesz potentials. Namely, it is shown that the two systems are equivalent under
some appropriate conditions. Then a sharp criterion for the existence and
non-existence of positive solutions is determined for both differential and
integral versions of a Hardy--Sobolev type system with variable coefficients.
In the constant coefficient case, Liouville type theorems for positive radial
solutions are also established using radial decay estimates and Pohozaev type
identities in integral form.
