It is well-known (see, for instance, [1], [2] and the references therein) that many integrable Hamiltonian systems arising in mechanics, geometry and mathematical physics are in fact bi-Hamiltonian. Recall that two Poisson brackets are called compatible if any their linear combination with constant coecients is also a Poisson bracket and that a dynamical system is called bi-Hamiltonian if it is Hamiltonian with respect to a pair of compatible Poisson brackets and all their non-trivial linear combinations.
In [3] and [4] several new methods were described that allow to construct first integrals using the bi-Hamiltonian structure of a system and even prove integrability for many important classes of bi-Hamiltonian systems. Since it is interesting to nd out if there are any other ways to naturally associate a set of commuting functions to a pair of compatible Poisson brackets, we will study the following question: if there are any integrable distributions that can be described only in terms of the bi-Hamiltonian structure itself?
In the talk we discuss the local structure of bi-Hamiltonian systems, including the classical Jordan-Kronecker theorem about the local structure of two bilinear form on a nite-dimensional vector space (see, for example, [5] and the references therein) and Turiel's theorem about the local structure of compatible symplectic structures in a neighborhood of a regular point (see [6]), and study the integrability of distributions which are invariant with respect to the group of local automorphisms for a pair of compatible Poisson brackets.

[1] Bolsinov, A. V., Oshemkov, A. A. Bi-Hamiltonian Structures and Singularities of Integrable Systems, Regular and Chaotic Dynamics, 2009, Vol. 14, no. 4-5, pp. 431-454.

[2] Borisov, A. V., Mamaev, I. S. Modern Methods of the Theory of Integrable Systems, Moscow - Izhevsk: Institute of Computer Science, 2003, 296 p.

[3] Magri, F. A Simple Model of the Integrable Hamiltonian Equation, J.Math. Phys., 1978, vol. 19, no. 5, pp. 1156-1162.

[4] Mishchenko, A. S. and Fomenko, A. T., Euler Equations on Finite-Dimensional Lie Groups, Izv. Akad. Nauk SSSR Ser. Mat., 42:2 (1978), 396-415.

[5] Kozlov I. K., An Elementary Proof of the JordanKronecker Theorem, Mat. Zametki, 94:6 (2013), 857-870.

[6] Turiel F. J., Classication locale simultanee de deux formes symplectiques compatibles, Manuscripta Math., 1994, vol. 82, no. 1, pp. 349-362.