Existence of Symmetric Pure Strategy Equilibrium
There is no guarantee that symmetric games have symmetric equilibria.
See this paper for concrete examples. 
There is also no guarantee that symmetric games have pure-strategy equilibria. For example,  the following game is symmetric and has no pure-strategy equilibria 
$$
\begin{array}{cccc}
&amp; \mathrm{a} &amp; \mathrm{b} &amp; \mathrm{c}\\
\mathrm{a} &amp; 1,1 &amp; 0,2 &amp; 5,0\\
\mathrm{b} &amp; 2,0 &amp; 3,3 &amp; 0,4\\
\mathrm{c} &amp; 0,5 &amp; 4,0 &amp; 1,1\\
\end{array}
$$
There are two special kinds of games that are known to always have pure strategy equilibria: supermodular games and potential games.