Non-degeneracy for the critical Lane-Emden system

We prove the non-degeneracy for the critical Lane-Emden system

$-\Delta U = V^p,\quad -\Delta V = U^q,\quad U, V > 0$$\quad \text {in } \mathbb{R}^N$

for all $N \ge 3$ and $p,q > 0$ such that $\frac {1}{p+1} + \frac {1}{q+1} = \frac {N-2}{N}$. We show that all solutions to the linearized system around a ground state must arise from the symmetries of the critical Lane-Emden system provided that they belong to the corresponding energy space or they tend to zero at infinity.