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 equat ions admitting 2 x 2 Lax matrices are 2N + k(0). It is known that N + k(0) pairs of canonical separated variables for their separation of variables ca n be introduced directly via their Lax matrices. Here we propose a new meth od to introduce the additional N pairs of canonical separated variables and N additional separated equations. The Jacobi inversion problems for high-o rder binary constrained flows and for soliton equations are also establishe d. This new method can be applied to all high-order binary constrained flow s admitting 2 x 2 Lax matrices.