  We investigate existence and qualitative properties of globally defined and
positive radial solutions of the Lane-Emden system, posed on a Cartan-Hadamard
model manifold $ \mathbb{M}^n $. We prove that, for critical or supercritical
exponents, there exists at least a one-parameter family of such solutions.
Depending on the stochastic completeness or incompleteness of $ \mathbb{M}^n $,
we show that the existence region stays one dimensional in the former case,
whereas it becomes two dimensional in the latter. Then, we study the
asymptotics at infinity of solutions, which again exhibit a dichotomous
behavior between the stochastically complete (where both components are forced
to vanish) and incomplete cases. Finally, we prove a rigidity result for
finite-energy solutions, showing that they exist if and only if $ \mathbb{M}^n
$ is isometric to $ \mathbb{R}^n $.
