Lax connection and conserved quantities of quadratic mean field games

Mean Field Game is a rather new field initially developed in applied mathematics and engineering order to deal with the dynamics of large number controlled agents or objects interaction. For class these models, there exists deep relationship between associated system equations non linear Schr\"odinger equation, which allows get insights on structure their solutions. In this work, we related aspects integrability for such systems, exhibiting some cases full hierarchy conserved quantities, bringing questions arise specific context.