Liouvillian Rst Integrals of Homogeneous Polynomial 3-dimensional Vector Elds

Given a 3-dimensional vector eld V with coordinates V x , V y and V z that are homogeneous polynomials in the ring kx; y; z], we give a necessary and suucient condition for the existence of a Liouvillian rst integral of V , which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in ring kx; y; z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaa and orthogonal to the vector eld V. Thus, our theorem links the existence of an object that belongs to some level of an extension tower with the existence of objects deened by means of the base diierential ring kx; y; z]. A self-contained proof of this result is given in the vocabulary of diierential algebra. This method of nding rst integrals in a given class of functions is an extension of the compatibility method introduced by J.-M. Strelcyn and S. Wojciechowski; and an old method of Darboux is a special case of it. We discuss all these relations and argue for the practical interest of our characterization despite an old open algorithmic problem.