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Serrin proved that, given a smooth bounded domain $\\Omega \\subset \\mathbb{R}^N$ and $u$ a positive solution of the problem:\n  \\begin{equation*}\n  \\begin{array}{ll}\n  -\\Delta u = f(u) &\\mbox{in $\\Omega$, }\n  u =0 &\\mbox{on $\\partial\\Omega$, }\n  \\partial_{\\nu} u =\\mbox{constant} &\\mbox{on $\\partial\\Omega$, }\n  \\end{array} \\end{equation*} then $\\Omega$ is necessarily a ball and $u$ is radially symmetric. In this paper we prove that the positivity of $u$ is necessary in that symmetry result. 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Serrin proved that, given a smooth bounded domain $\\Omega \\subset \\mathbb{R}^N$ and $u$ a positive solution of the problem:\n  \\begin{equation*}\n  \\begin{array}{ll}\n  -\\Delta u = f(u) &\\mbox{in $\\Omega$, }\n  u =0 &\\mbox{on $\\partial\\Omega$, }\n  \\partial_{\\nu} u =\\mbox{constant} &\\mbox{on $\\partial\\Omega$, }\n  \\end{array} \\end{equation*} then $\\Omega$ is necessarily a ball and $u$ is radially symmetric. In this paper we prove that the positivity of $u$ is necessary in that symmetry result. 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