Summary: We consider the fully nonlinear integral systems involving Wolff potentials:
After modifying and refining our techniques on the method of moving places in integral forms, we obtain radial symmetry and monotonicity for the positive solutions to system (1). This system includes many known systems as special cases, in particular, when and , system (1) reduces to
The solutions of (2) are critical points of the functional associated with the well-known Hardy-Littlewood-Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic partial differential equations
which comprises the well-known Lane-Emden system and Yamabe equation.