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This work is devoted to the study of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many solutions of a related nonlinear eigenvalue problem. Applying an abstract minimax theorem, we obtain a solution of the quasilinear system −Δpu=Fu(x, u, v), −Δqv=F v(x, u, v), under conditions involving the first and the second eigenvalues.