Existence of Symmetric Pure Strategy Equilibrium I have 2 symmetric players $A$ and $B$. 

Each of them has 2 decision variables $x_i\in[0, \beta]$ and $y_i\in[0,1]$, where $i\in\{A,B\}$. 

Their payoff functions are symmetric, i.e., if you swap the label $A$ and $B$, you get the other player's payoff function. And the payoff function is continuous in both $x$ and $y$. 

My question is: for this type of game, does it always have symmetric pure strategy equilibrium? In general, what are the conditions to guarantee the 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.