Noncommutative integration

The problem listed above are particular cases of the general noncommutative integration problem which b, in fact, traced back to Cartan and has been analyzed from different points of view by Fomenko and Trofimov in the survey [188], and Marsden and Weinstein in [78]. 

The noncommutative integration method described in 1 was proposed by Fomenko and Mishchenko and then developed by Brailov and extended to the case of a larger collection of subalgebras in the Lie algebra C (M). In this subsection, we briefly present the result obtained by Brailov. 

It follows from Proposition 3.3.1 that the definition of complete noncommutative integrability given above may be formulated in the form close to formula (2). Kamely, a Hamiltonian system on a symplectic manifold M with the Lie algebra G of integrals is completely integrable in the noncommutative sense if... 

In the resonant case, integral trajectories are everywhere dense on tori of smaller dimension. We recall that precisely such a situation characterizes those Hamiltonian systems which admit noncommutative integration (see above). 

Brailov, A. V. Complete integrability of some geodesic flows and integrable systems with noncommuting integrals. DokL Akad. Nauk SSSR 271 (1983), No. 2, 273-276. 

Brailov, A. V. Complete integrability with noncommuting integrals of several Euler equations. In Application of Topology in Modern Analysis. Voronezh, Voronezh Univ. Press, (1985), 22-41. 

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