  We study the following Lane-Emden system \[ -\Delta u=|v|^{q-1}v \quad \text{
in } \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \text{ in } \Omega, \qquad
u_\nu=v_\nu=0 \quad \text{ on } \partial \Omega, \] with $\Omega$ a bounded
regular domain of $\mathbb{R}^N$, $N \ge 4$, and exponents $p, q$ belonging to
the so-called critical hyperbola $1/(p+1)+1/(q+1)=(N-2)/N$. We show that, under
suitable conditions on $p, q$, least-energy (sign-changing) solutions exist,
and they are classical. In the proof we exploit a dual variational formulation
which allows to deal with the strong indefinite character of the problem. We
establish a compactness condition which is based on a new Cherrier type
inequality. We then prove such condition by using as test functions the
solutions to the system in the whole space and performing delicate asymptotic
estimates. If $N \ge 5$, $p=1$, the system above reduces to a biharmonic
equation, for which we also prove existence of least-energy solutions. Finally,
we prove some partial symmetry and symmetry-breaking results in the case
$\Omega$ is a ball or an annulus.
