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For nonlinear difference equations, it is difficult to obtain analytic solutions, especially when all the eigenvalues of the equation are of absolute value 1. We consider a second order nonlinear difference equation which can be transformed into the following simultaneous system of nonlinear difference equations: ⎧ x(t+1) = X(x(t),y(t)) ⎨ ⎩ y(t+1) = Y(x(t), y(t)) where $X(x,y)=\lambda ₁x+\mu y+{\sum }_{i+j\ge 2}{c}_{ij}{x}^i{y}^j$, $Y(x,y)=\lambda ₂y+{\sum }_{i+j\ge 2}{d}_{ij}{x}^i{y}^j$ satisfy some conditions. For these equations, we have obtained analytic solutions in the cases "|λ₁| ≠ 1 or |λ₂| ≠ 1" or "μ...