Let Ω ⊂ R n be a bounded smooth open set. We prove that the singular set of any extremal solution of the system −∆u = µe v , −∆v = λe u in Ω, with u = v = 0 on ∂Ω, µ, λ ≥ 0, has Hausdorff dimension at most n − 10. 2010 Mathematics Subject Classification. Primary: 35G30, 35B65; Secondary: 35P30.