Non-degeneracy for the critical Lane--Emden system

We prove the non-degeneracy for the critical Lane-Emden system \displaystyle -\Delta U = V^p,\quad -\Delta V = U^q,\quad U, V > 0\displaystyle \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.