Using pedestrian algebraic method, in the case of a coupled system of n linear first order differential equations (Δ0−iφΛ −Q) ψ=0, Λ= (δijλi), Δ0 = (δijμi(x)(∂/∂x)), Q= (qij), we deduce inversionlike integral equations (from which we can construct a class of potentials without introducing the data). Assuming, like in the Zakharov–Shabat theory, that the kernels of the linear integral inversion equation satisfy linear partial differential equations, we deduce, for the solution of the integral equation, the corresponding nonlinear partial differential equation. We show that the validity of the whole formalism depends mainly on the set of {λi} of the associated differential linear system. We discuss fully the different possibilities which occur depending on different sets of {λi}.