Algorithms for special integrals of ordinary differential equations

We give new, conceptually simple procedures for calculating special integrals of polynomial type (also known as Darboux polynomials, algebraic invariant curves, or eigenpolynomials), for ordinary differential equations. In principle, the method requires only that the given ordinary differential equation be itself of polynomial type of degree one and any order. The method is algorithmic, is suited to the use of computer algebra, and does not involve solving large nonlinear algebraic systems. To illustrate the method, special integrals of the second, fourth and sixth Painlevé equations, and a third-order ordinary differential equation of Painlevé type are investigated. We prove that for the second Painlevé equation, the known special integrals are the only ones possible.