An efficient way to determine Liouvillian first integrals of rational second order ordinary differential equations

The Lie and Darboux methods to obtain Liouvillian first integrals of rational second order ordinary differential equations (rational 2ODEs) are very powerful. Nevertheless, there are cases where these procedures may encounter difficulties. Namely, when the Darboux polynomials present in the integrating factor of the 2ODE have a very high degree and/or when the 2ODE does not admit point symmetries. In [3], [4] we developed a method (S-function method) that is successful in treating certain classes of rational second order ordinary differential equations that are particularly ‘resistant’ to canonical Lie methods and to Darbouxian approaches. However, although determining the S-function is in general much more efficient than determining Darboux polynomials or nonlocal symmetries, for very complicated rational 2ODEs, even finding the S-function itself can be quite hard. In this work we present a simple way of (in almost all cases) computing the S-function with a very efficient procedure.