Noncommutative integrable systems pertain to analogues of integrable systems in which the coordinates and momenta are formal noncommuting variables.
Pavel Etingof, Israel Gelfand, Vladimir Retakh, Nonabelian integrable systems, quasideterminants, and Marchenko lemma, Math. Res. Lett. 5 (1998), no. 1-2, 1–12, MR99h:58083