We study the supersymmetric vacua of the Veneziano–Wosiek model in sectors with fermion number F = 2, 4 at finite 't Hooft coupling λ. We prove that for F = 2 there are two zero energy vacua for λ>λ_c = 1 and none otherwise. We give the analytical expressions of both vacua. One of them was previously known, the second one is obtained by solving the cohomology of the supersymmetric charges. At F = 4 we compute the would-be supersymmetric vacua at high order in the the strong coupling expansion and provide strong support to the conclusion that λ = 1 is a critical point in this sector too. It separates a strong coupling phase with two symmetric vacua from a weak coupling phase with positive spectrum.