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Suppose the equation x + g(x) = -λ₁x + λ₂f where f is a scalar function which is 2π-periodic, λ₁, λ₂ are real parameters, xg(x) > 0 for x ≠ 0. The initial problem is to Characterize the existence and the number of 2π-periodic solutions of (1) which lie in a neighborhood of a 2π-periodic orbit of the degenerated equation x + g(x) = 0 (2) whose orbit in the (x, x) - space encircles the origin. The Liapunov-Schmidt reduction method is applied to obtain the bifurcation equations. The results are then obtained by successive use of the Implict Function theorenm,. We also characterize the existence and the number of 4π-periodic solutions of (1) which lie in a neighborhood of a 2π-periodic orbit of (2).