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Many processes are mathematically simulated by systems of discrete equations with quadratic right‐hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalue λ = 1 of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.