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ґ V. Dragovic Mathematical Institute SANU Belgrade, Serbia vladad@mi.sanu.ac.yu ґ B. Gajic Mathematical Institute SANU Belgrade, Serbia gajab@mi.sanu.ac.yu ґ B. Jovanovic Mathematical Institute SANU Belgrade, Serbia bozaj@mi.sanu.ac.yu

Complete commutative algebras on T S O(n)/S O(k1 ) в S O(k2 ) в · · · в S O(kr )
Mishchenko and Fomenko stated the conjecture that non-commutative integrable systems are integrable in the usual commutative sense by means of integrals that belong to the same functional class as the original non-commutative algebra F of integrals. If F is a finitedimensional Lie algebra the conjecture has been proved recently by Sadetov [4] (see also [5]). In the smooth category, the answer is positive in the infinitely dimensional case as well [2]. The important class of analytical non-commutative integrable systems are geodesic flows of normal metric on homogeneous spaces G/H of compact Lie groups. In this case, the conjecture reduses to the construction of complete commutative algebras within G-invariant functions on T (G/H ) [2]. For example if (G, H ) is a spherical pair, the algebra of G-invariant functions is already commutative. In many examples, such as Stiefel manifolds, flag manifolds, orbits of the adjoint actions, commutative algebras are obtained (see [2, 3]), but the general problem is still unsolved. Following [1], by using the shift-argument method, we present the construction of complete commutative algebras for homogeneous spaces S O(n)/S O(k1 ) в · · · в S O(kr ) for any choice of k1 , . . . , kr . References [1] Bolsinov, A. V.: Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Izv. Acad. Nauk SSSR, Ser. matem. 55, no.1, 68-92 (1991) (Russian); English translation: Math. USSR-Izv. 38, no.1, 69-90 (1992) [2] Bolsinov, A. V., Jovanovic, B.: Integrable geodesic flows on Riemannian manifolds: Construction and Obstructions; In: Contemporary Geometry and Related Topics (Eds. Bokan N., Djoric M., Rakic Z., Fomenko A. T., Wess J.), World Scientific, 2004, pp. 57-103., arXiv:math-ph/0307015 [3] Mykytyuk, I. V. and Panasyuk A.: Bi-Poisson structures and integrability of geodesic flows on homogeneous spaces. Transformation Groups 9, no. 3, 289-308 (2004) [4] Sadetov, S. T.: A proof of the Mishchenko-Fomenko conjecture (1981). Dokl. Akad. Nauk 397, no. 6, 751­754 (2004) (Russian) [5] Vinberg, E. B., Yakimova, O. S.: Complete families of commuting functions for coisotropic Hamiltonian actions, arXiv: math.SG/0511498.

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