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By means of the nonlinearization technique, a Bargmann constraint associated with a new discrete 4 × 4 matrix eigenvalue problem is proposed, and a new symplectic map of the Bargmann type is obtained through binary nonlinearization of the discrete eigenvalue problem and its adjoint one. Moreover, the generating function of integrals of motion is obtained, by which the symplectic map is further proved to be completely integrable in the Liouville sense. Finally, the involutive representation of solutions are given.

Based on the nonlinearization technique, a binary Bargmann symmetry constraint associated with a new discrete 3 × 3 matrix eigenvalue problem, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals, is proposed. A new symplectic map of the Bargmann type is obtained through binary nonlinearization of the discrete eigenvalue problem and its adjoint one. The generating function of integrals of motion is obtained, by which the symplectic map is further proved to be completely integrable in the Liouville sense. c ⃝2015 All rights reserved.

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