On the remarkable values of the rational first integrals of polynomial vector fields

The remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by Poincaré. He was mainly interested in their algebraic aspects. Here we are interested in their dynamic aspects; i.e. how they contribute to the phase portrait of the system, to its separatrices, to its singular points, etc. The relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor.