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This work advances the understanding of oscillator Ising machines (OIMs) as a nonlinear dynamic system for solving computationally hard problems. Specifically, we classify the infinite number of all possible equilibrium points of an OIM, including non-0/π ones, into three types based on their structural stability properties. We then employ the stability analysis techniques from control theory to analyze the stability property of all possible equilibrium points and obtain the necessary and sufficient condition for their stability. As a result of these analytical results, we establish, for the first time, the threshold of the binarization in terms of the coupling strength and strength of the second harmonic signal. Furthermore, we provide an estimate of the domain of attraction of each asymptotically stable equilibrium point by employing the Lyapunov stability theory. Finally, we illustrate our theoretical conclusions by numerical simulation.