A connection between nonlinear evolution equations and ordinary differential equations of P-type. I

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Abstract
We develop here two aspects of the connection between nonlinear partial differential equations solvable by inverse scattering transforms and nonlinear ordinary differential equations (ODE) of P‐type (i.e., no movable critical points). The first is a proof that no solution of an ODE, obtained by solving a linear integral equation of a certain kind, can have any movable critical points. The second is an algorithm to test whether a given ODE satisfies necessary conditions to be of P‐type. Often, the algorithm can be used to test whether or not a given nonlinear evolution equation may be completely integrable.