  We study existence, nonexistence, and uniqueness of positive radial solutions
for a class of nonlinear systems driven by Pucci extremal operators under a
Lane-Emden coupling configuration. Our results are based on the analysis of the
associated quadratic dynamical system and energy methods. For both regular and
exterior domain radial solutions we obtain new regions of existence and
nonexistence. Besides, we show an exclusion principle for regular solutions,
either in $\mathbb{R}^N$ or in a ball, by exploiting the uniqueness of
trajectories produced by the flow.
  In particular, for the standard Lane-Emden system involving the Laplacian
operator, we prove that the critical hyperbola of regular radial positive
solutions is also the threshold for existence and nonexistence of radial
exterior domain solutions with Neumann boundary condition. As a byproduct,
singular solutions with fast decay at infinity are also found.
