Exact Linearization of a Painlevé Transcendent

There is a connection between nonlinear partial differential equations that can be solved by the inverse scattering transform and nonlinear ordinary differential equations without movable critical points (e.g., Painlev\'e transcendents). We exploit this connection to reduce the second equation of Painlev\'e to a linear integral equation. We also describe a class of nonlinear ordinary differential equations that can be exactly linearized by this method.