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Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds Förderungs- und Betriebs GmbH Schwerpunkte:In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that