  The two-players $N$ strategies games quantized according to the
Eisert-Lewenstein-Wilkens scheme (Phys. Rev. Lett. 83 (1999), 3077) are
considered. Group theoretical methods are applied to the problem of finding a
general form of gate operators (entanglers) under the assumption that the set
of classical pure strategies is contained in the set of pure quantum ones. The
role of the stability group of the initial state of the game is stressed. As an
example, it is shown that the maximally entangled games do not admit nontrivial
pure Nash strategies. The general arguments are supported by explicit
computations performed in the three strategies case.
