Summary: We consider the system of integral equations in :
with . Under some integrability conditions, we obtain radial symmetry of positive solutions by using the method of moving planes in integral forms. In the special case when , we show that the integral system is equivalent to the elliptic PDE system
in . Our symmetry result, together with non-existence of radial solutions by E. Mitidieri [Commun. Partial Differ. Equations 18, No. 1–2, 125–151 (1993; Zbl 0816.35027], implies that, under our integrability conditions, the PDE system possesses no positive solution in the subcritical case. This partially solved the well-known Lane-Emden conjecture.