Integrable decompositions of a symmetric matrix Kaup–Newell equation and a symmetric matrix derivative nonlinear Schrödinger equation

It is shown that the reduction conditions in the spectral problems of soliton equations can be replaced by more equations of eigenfunctions. Based on this, the method of nonlinearization of spectral problem is extended to soliton equations with reduction conditions. Integrable decompositions of a symmetric matrix Kaup–Newell equation and a symmetric matrix derivative nonlinear Schrödinger equation are constructed.