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In this note we generalize and extend the results obtained in a recent work of Zhang and Cai \\cite{ZC,ZC2}. More precisely, we prove that for each $n\\geq 4$ and rational numbers $a, b$ with $ab\\neq 0$, the system of diophantine equations \\begin{equation*}\n  \\sigma_{1}(x_{1},\\ldots, x_{n})=a, \\quad \\sigma_{n}(x_{1},\\ldots, x_{n})=b, \\end{equation*} has infinitely many solutions depending on $n-3$ free parameters. 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