  The paper is devoted to the first-order mean field game system in the case
when the distribution of players can contain atoms. The proposed definition of
a generalized solution is based on the minimax approach to the Hamilton-Jacobi
equation. We prove the existence of the generalized (minimax) solution of the
mean filed game system using the Nash equilibrium in the auxiliary differential
game with infinitely many identical players. We show that the minimax solution
of the original system provide the $\varepsilon$-Nash equilibrium in the
differential game with finite number of players.
