Parametric Painlevé equations

The parametric Painlevé equations are those ODEs whose general solutions can be presented in the parametric form in terms of the Painlevé functions. Most of these ODEs do not possess the Painlevé property. By considering similarity solutions of the short pulse equation and its decoupled generalization we derive a non-trivial example of the parametric Painlevé equation related with the third Painlevé equation. We also discuss some analytic properties of this equation describing the structure of movable singularities.