Is my ODE a Painlevé equation in disguise?

Painlevé equations belong to the class y''+a1 y'3+3a2 y'2+3a3 y'+a4 = 0, where ai = ai(x, y). This class of equations is invariant under the general point transformation x = φ(X, Y ), y = Ψ(X, Y ) and it is therefore very difficult to find out whether two equations in this class are related. We describe R. Liouville's theory of invariants that can be used to construct invariant characteristic expressions (syzygies), and in particular present such a characterization for Painlevé equations I-IV. Copyright © 2002 by J Hietarinta and V Dryuma.