Reduction of integrable planar polynomial differential systems

The integrability problem consists in finding the class of functions, a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit a Darboux, elementary, Liouvillian or Weierstrass first integral. The reduction problem of an integrable planar system consists in finding the class of functions, a map that reduces the original system (transforms into a simple system or equation) must belong to. We identify the class of functions of this map for polynomial, rational, Darboux, elementary, Liouvillian and Weierstrass integrable systems.