  In this paper, we consider the critical Lane-Emden system \begin{align*}
\begin{cases} -\Delta u=K_1(y)v^p,\quad y\in \mathbb{R}^N,&\\ -\Delta
v=K_2(y)u^q,\quad y\in \mathbb{R}^N,&\\ u,v>0, \end{cases} \end{align*} where
$N\geq 5$, $p,q\in (1,\infty)$ with
$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$, $K_1(y)$ and $K_2(y)$ are positive
radial potentials. Under suitable conditions on $K_1(y)$ and $K_2(y)$, we
construct a new family of solutions to this system, which are centred at points
lying on the top and the bottom circles of a cylinder.
