Structures on manifolds and algebraic integrability of dynamical systems

Abstract

During recent decades it has been stated (Golod, 1984), (Golod, 1982), (Lebedev et al., 1979a), (Reymann et al., 1980), (Takhtadjian et al., 1987), (Adler, 1979), (Jacob et al., 1980), (Kazhdan et al., 1978), (Kupershmidt et al., 1983), (Symes, 1980), (Blaszak, 1995a), (Blaszak, 1993c), (Blaszak, 1994) that many dynamical systems of classical physics and mechanics are endowed with the symplectic structure associated with the Poisson bracket. In many such cases the structure of the Poisson bracket is canonical and is given upon the dual space of the corresponding Lie algebra of symmetries, being added in some cases (Reymann et al., 1980), (Takhtadjian et al., 1987), (Kupershmidt et al., 1983) with a 2-cocycle, and sometimes having a gauge nature. These observations give rise to a deep group-theoretical interpretation of Poisson structures for many integrable dynamical systems of mathematical physics.