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

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.