  Let $\sigma_{i}(x_{1},\ldots, x_{n})=\sum_{1\leq k_{1}<k_{2}<\ldots
<k_{i}\leq n}x_{k_{1}}\ldots x_{k_{i}}$ be the $i$-th elementary symmetric
polynomial. 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*}
  \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. A similar result is proved for the system \begin{equation*}
  \sigma_{i}(x_{1},\ldots, x_{n})=a, \quad \sigma_{n}(x_{1},\ldots, x_{n})=b,
\end{equation*} with $n\geq 4$ and $2\leq i< n$. Here, $a, b$ are rational
numbers with $b\neq 0$.
  We also give some results concerning the general system of the form
\begin{equation*}
  \sigma_{i}(x_{1},\ldots, x_{n})=a, \quad \sigma_{j}(x_{1},\ldots, x_{n})=b,
\end{equation*} with suitably chosen rational values of $a, b$ and $i<j<n$.
Finally, we present some remarks on the systems involving three different
symmetric polynomials.
