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The Bargmann Constraint and Integrable System for a Second-Order Spectral Problem

Abstract In this article, the Bargmann system related to the spectral problem (∂2+q∂+∂q+r)φ=λφ+λφx is discussed. By the Euler–Lagrange equations and the Legendre transformations, a suitable Jacobi–Ostrogradsky coordinate system is obtained. So the Lax pairs of the aforementioned spectral problem are nonlinearised. A new kind of finite-dimensional Hamilton system is generated. Moreover, the involutive solutions of the evolution equations for the infinite-dimensional soliton system are derived.