We prove the non-degeneracy for the critical Lane–Emden system
−ΔU = V^p, −ΔV = U^q, U,V >0 in R^N
for all N ≥ 3 and p, q > 0 such that 1/(p+1) + 1/(q+1) = (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
