A new method to introduce additional separated variables for high-order binary constrained flows

Degrees of freedom for high-order binary constrained flows of soliton equations admitting 2 × 2 Lax matrices are 2N +k 0 . It is known that N +k 0 pairs of canonical separated variables for their separation of variables can be introduced directly via their Lax matrices. Here we propose a new method to introduce the additional N pairs of canonical separated variables and N additional separated equations. The Jacobi inversion problems for high-order binary constrained flows and for soliton equations are also established. This new method can be applied to all high-order binary constrained flows admitting 2 × 2 Lax matrices.