The Third Order Ordinary Differential Equations Admitting the Painlev´e I and II as First Integrals

Third order ordinary differential equations admitting the Painlev´e equations PI and PII as first integrals are completely characterized. This is done by determining necessary and sufficient conditions for these equations to belong to the equivalence class of certain canonical equations. This characterization also includes equations admitting first integrals which are equivalent to PI and PII. The general form of these equations is given. In some cases we determine the transformation taking the equation into a canonical one.