  Mean field games equations are examined for conservation laws. The system of
mean field games equations consists of two partial differential equations: the
Hamilton-Jacobi-Bellman equation for the value function and the forward
Kolmogorov equation for the probability density. For separable Hamiltonians,
this system has a variational structure, i.e., the equations of the system are
Euler-Lagrange equations for some Lagrangian functions. Therefore, one can use
the Noether theorem to derive the conservation laws using variational and
divergence symmetries. In order to find such symmetries, we find symmetries of
the PDE system and select variational and divergence ones. The paper considers
separable, state-independent Hamiltonians in one-dimensional state space. It
examines the most general form of the mean field games system for symmetries
and conservation laws and identifies particular cases of the system which lead
to additional symmetries and conservation laws.
