If the players are symmetric and the core is nonempty, then $x_i = v(N)/n$ for all $i$ is a Core element
The center allocation $z_{i}:=(v(N)/|N|)$ for all $i \in N$ is in the core of the symmetric game $v$, whenever $z(S) = \sum_{i \in S} z_{i} \ge v(S) $ holds for all $S \subseteq N$. Now, if $v(N)/|N| \ge v(S)/|S|$ for all $S \subseteq N$, then it holds $z(S) = \sum_{i \in S} z_{i} = |S|\cdot v(N)/|N| \ge v(S)$. Thus, $\mathbf{z} \in C(v)$.  

We observe that for a symmetric game $v$, the core is non-empty if $v(N)/|N| \ge v(S)/S$ holds for all $S \subseteq N$. Otherwise, the core is empty.