I guess, in order to answer this question, I need to write Payoff matrix. But I cannot write it. And then, I Will able to answer this question by myself. Thank you for helping.
(These are just studying-exercise questions. I Try to learn this topic by the help of the questions.)
The payoff matrix is huge in this game since there are so many possible strategies (all the different ways to distribute the $120$ soldiers to the six battlefields).
You just need to check the given configuration of armies. Is the response of $B$ to the $A$'s configuration a best response and vice versa. If so, this is an equilibrium.
EDIT: OK from the formulation of the question I understand that a tie means losing.
RE_EDIT: Sorry I didn't think this through. Of course $A$ can change distribution to for example: $(0,38, 22, 22, 21, 17)$ to have a win. So this isn't an equilibrium.
Sorry editing all the time but now I think I got it:
The equlibria are those where all the soldiers are divided equally to $3$ battlefields (then the other general cannot win the war and this always forces a tie). So the equlibria are (for example): $A$ plays $(40, 40, 40, 0, 0, 0)$ and it doesn't matter what $B$ plays. Of course you can change the places where the $40$'s are.
Also the ones where $B$ plays the $40$-strategy and it doesn't matter what $A$ plays are equlibria.
Hope I got this right now... :D.