  In this paper we consider lower order perturbations of the critical
Lane-Emden system posed on a bounded smooth domain $\Omega \subset
\mathbb{R}^N$, with $N \geq3$, inspired by the classical results of Brezis and
Nirenberg \cite{BrezisNirenberg1983}. We solve the problem of finding a
positive solution for all dimensions $N \geq 4$. For the critical dimension
$N=3$ we show a new phenomenon, not observed for scalar problems. Namely, there
are parts on the critical hyperbola where solutions exist for all
$1$-homogeneous or subcritical superlinear perturbations and parts where there
are no solutions for some of those perturbations.
