A nonconfocal involutive system and constrained flows associated with the MKdV − equationThe special quasiperiodic solution of the (2ϩ1)-dimensional Kadometsev-Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the well-known confocal involutive hierarchy associated with the nonlinearized Zakharov-Shabat eigenvalue problem. The explicit theta function solution of the Kadometsev-Petviashvili equation is obtained with the help of this separation technique. A generating function approach is introduced to prove the involutivity and the functional independence of the conserved integrals which are essential for the Liouville integrability.Hence c 1 ϭ¯ϭc NϪ1 ϭ0 since the coefficient determinant is equal to 1 by Eq. ͑4.6͒. Thus c 0 dF 0 ϭ0. And c 0 ϭ0 since dF 0 ϭϪ ͚ ͑ q k dp k ϩ p k dq k ͒ 0. 3955

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The H1 model in the Adler–Bobenko–Suris list, i.e. the lattice potential KdV equation and the closely related lattice KdV equation with Nijhoff’s discretization, are investigated. A new Lax pair of the H1 model is given, by which integrable symplectic maps are constructed through a non-linearization procedure. Resorting to these maps, finite genus solutions of the H1 model as well as the lattice KdV equation are calculated.

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The Darboux transformations (DTs) associated with the well-known ZS-AKNS spectral problem are regarded as discrete spectral problems. Two kinds of integrable symplectic maps are derived from them through nonlinearization procedures. As an application, they are used to calculate the finite genus solutions of some new integrable lattice models constructed from these DTs.