Conjecture relating the inverse scattering method to Painleve-type equations

The following conjecture is proposed: for any nonlinear partial differential equation or system solvable y the Inverse Scattering Transform method, and for any ansatz that reduces it to a nonlinear ordinary differential equation or system, the resulting equation or system is of Painleve type, which means that its solutions have no movable singularities other than poles. Several credibility arguments are offered to support this conjecture. Then an algorithm is described that gives a necessary condition for an equation or a system to be of Painleve type. As an auxiliary problem, it is shown that it is not necessary to go through the formalism of the direct scattering problem (Jost functions, etc.) to prove that the solution of the linear integral equation of Gel'fand-Levitan-Marchenko type is indeed a solution of the nonlinear differential equation it is supposed to solve. This is useful in connection with ordinary differential equations that might have no solutions with the properties necessary for the usual approach to make sense