The scaling reduction of the three-wave resonant system and the Painlev VI equation

The scaling invariant solutions of the three-wave resonant system in one spatial and one temporal dimension satisfy a system of three first-order nonlinear ordinary differential equations. These equations can be reduced to one second-order equation quadratic in the second derivative. This equation is outside the class of equations classified by Painlevé and his school. However, it is a special case of an equation recently found to be related via a one-to-one transformation to the Painlevé VI equation.