  The two-players N strategies games quantized according to the
Eisert-Lewenstein-Wilkens scheme [1] are considered. It is shown that in the
case of maximal entanglement no nontrivial pure Nash equilibrium exists. The
proof relies on simple geometric properties of "chiral" group $SU (N)\times
SU(N)$ and is based on considering the stability subgroup of the initial state
of the game. The explicit forms of neither the gate operator nor the payoff
matrix are necessary.
