I’m currently struggling with the following problem:

Given a generic quadratic function Q on g* (where g is a simple Lie algebra) which Poisson-commute with a Cartan subalgebra t of g, show that the Hamiltonian system of Q does not admit any other first integral exept the obvious ones (the ones which can be functionally generated . . . → Read More: non-integrability of hamiltonian systems on g*

I’m collecting in this page references directly related to the Gelfand-Cetlin system.

The list is ordered chronologically.

* Gelfand & Cetlin ? (around 1950)

* Thimm: Integrable geodesic flows on homology spheres, Ergodic Th. Dyn. Syst. 1 (1981), 595–517. (the first paper in which the integrable GC system appears ?)

* Guillemin & Sternberg: The Gelfand-Cetlin system and quantization . . . → Read More: References around Gelfand-Cetlin

A student of mine, Michael Ayoul, is currently writing up the non-Hamiltonian version of Morales-Ramis(-Simo) theorem, which says that the differential Galois group of the variational equation (of any order) of a meromorphic integrable  system along a solution must be Abelian.

He will then try to apply this result to various non-Hamiltonian (non-holonomous) systems to show . . . → Read More: Non-Hamiltonian version of Morales-Ramis theorem

I have a PhD student, Iman Allamiddine, to whom I gave the following problem: study the geometry/topology, in articular the singularities, of the Gelfand-Cetlin system.

It seems that these systems have singularities which are quite different from typical singularities of integrable Hamiltonian systems met in classical mechanics ?

We are collecting references on Gelfand-Cetlin systems, so you you . . . → Read More: Gelfand-Cetlin Systems