  In this paper, we study the nonexistence of positive supersolutions for the
following Lane-Emden system with inverse-square potentials
  \begin{equation}\label{0}
  \left\{ \begin{array}{lll}
  -\Delta u+\frac{\mu_1}{|x|^2} u= v^p \quad {\rm in}\ \,
\Omega\setminus\{0\},\\[2mm]
  -\Delta v+\frac{\mu_2}{|x|^2} v= u^q \quad {\rm in}\ \, \Omega\setminus\{0\}
  \end{array}
  \right.
  \end{equation} for suitable $p,q>0$, $\mu_1,\mu_2\geq -(N-2)^2/4$, where
$\Omega$ is a smooth bounded domain containing the origin in $\mathbb{R}^N$
with $N\geq 3$. Precisely, we provide sharp supercritical regions of $(p,q)$
for the nonexistence of positive supersolutions to system (\ref{0}) in the
cases $-(N-2)^2/4\leq \mu_1,\mu_2<0$ and $-(N-2)^2/4\leq \mu_1<0\leq \mu_2$.
Due to the negative coefficients $\mu_1,\mu_2$ of the inverse-square
potentials, an initial blowing-up at the origin could be derived and an
iteration procedure could be applied in the supercritical case to improve the
blowing-up rate until the nonlinearities are not admissible in some weighted
$L^1$ spaces. In the subcritical case, we prove the existence of positive
supersolutions for system (\ref{s 1.1}) by specific radially symmetric
functions.
