The relation between potential games and identical interest commutative utility functions

Question

Assume a graphical game, where the utility of $i$-th player only depends on its neighboring players, i.e. $$ u_i(x_i,x_{-i})=u_i(x_i,x_{N(i,1)},...,x_{N(i,M_i)})$$ where: $N(i,j)$ denotes the $j$-th neighboring player to player $i$, and $M_i$ represents the number of players in the neighborhood of player $i$.
We call neighborhood based utility functions commutative if for all $i$: $$u_i(x_i,x_{-i})=u_i(y) $$ for any permutation $y$ of $x_i,x_{N(i,1)},...,x_{N(i,M_i)}$.
Consider that all players have identical commutative utility functions. Is this game a potential game, or is there any relations between the two?