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We prove that if ${\\rm Spc}\\,{(\\phi)}\\subset\\R$ or ${\\rm Spc}\\,(\\phi)\\cap [1,1+\\epsilon)=\\emptyset$ for some $\\epsilon>0$ then $\\phi$ is globally $C^1$ conjugate to the linear involution $D\\phi(0)$ via the conjugacy $h=(I+D\\phi(0)\\phi)/2$, where $I:\\R^2\\to\\R^2$ is the identity map. 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