An algorithm to convert integrable third-order ODEs to regular higher-order equations near any movable singularities

A general property for differential equations passing the Painlevé test is presented. Suppose a solution u of a potentially integrable equation has poles near movable singularities, then u⁻¹ should be an analytic function near movable singularities of u. If we believe integrable equations behave nicely, then we should expect u⁻¹ to satisfy some regular equation near movable singularities of u. In this paper, we formulate an algorithm of converting any general third-order ordinary differential equations to regular higher-order equations near movable singularities. For demonstration, the algorithm is used to find the regular equation of time-independent Korteweg-de Vries equation. Copyright © 2020 Hikari Ltd.