  We extend the methods from Nurbekyan, Saude "Fourier approximation methods
for first-order nonlocal mean-field games" [Port. Math. 75 (2018), no. 3-4] and
Liu, Jacobs, Li, Nurbekyan, Osher "Computational methods for nonlocal mean
field games with applications" [arXiv:2004.12210] to a class of non-potential
mean-field game (MFG) systems with mixed couplings. Up to now, splitting
methods have been applied to potential MFG systems that can be cast as
convex-concave saddle-point problems. Here, we show that a class of
non-potential MFG can be cast as primal-dual pairs of monotone inclusions and
solved via extensions of convex optimization algorithms such as the primal-dual
hybrid gradient (PDHG) algorithm. A critical feature of our approach is in
considering dual variables of nonlocal couplings in Fourier or feature spaces.
