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The book consists of three chapters. In chapter I, we introduce some notions and definitions for basic concepts of the theory of integrable bi-Hamiltonian systems. Several open problems related to our main results are also mentioned in this part. In chapter II. We applied the so-called Jordan-Kronecker decomposition theorem to study algebraic properties of the pencil generated by two constant compatible Poisson structures on a vector space. In particular, we study the linear automorphism group that preserves the pencil. In classical symplectic geometry, many fundamental results are based on the symplectic group, which preserves the symplectic structure. Therefore in the theory of bi-Hamiltonian structures, we hope the linear automorphism group also plays a fundamental role. In chapter III, We describe the Lie group of linear automorphisms of the pencil, and obtain an explicit formula for the dimension of the Lie group and discuss some other algebraic properties such as solvability and Levi-Malcev decomposition.

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Pumei Zhang works as an Operational Research and Ordinary Differential Equations instructor at China University of Political Science and Law. She holds at BSc and a PhD in Mathematics at Loughborough University. In addition, she is interested and working on some applied mathematic project related to operations research and open digital innovation.

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