Abstract
Abstract A new Neumann type finite-dimensional Hamilton system (FDHS) is presented by means of the nonlinearization of Lax pairs (NLP). Its Lax representation is deduced from the auxiliary spectral problem. Furthermore, it is displayed that the Lax representation satisfies a dynamic r -matrix relation in the Dirac bracket. Additionally the Lax representation and r -matrix for the Bargmann type FDHS are also given in the standard Poisson bracket. Consequently, the integrability in the Liouville sense of the resulting FDHSs is completed in the framework of r -matrix theory.
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