I never heard about this kind of game. So to summarize (if I understood correctly), this would be a variation of the classical 8 queens problem [0] but on a wider board. Very interesting !
It is a little more complicated because the queens move around from turn to turn, they shoot "arrows" to block each other's movements, and they are not forbidden from being in a position to attack one another in the chess sense. Those aren't features of the traditional eight queens problem. (I haven't played the game, I just read the rules online.)