Liouvillian first integrals of homogeneouspolynomial 3-dimensional vector fields
Given a 3-dimensional vector field V with coordinates $V_x$, $V_y$ and $V_z$ that are homogeneous polynomials in the ring k[x,y,z], we give a necessary and sufficient condition for the existence of a Liouvillian first integral of V which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in the ring k[x,y,z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaff and orthogonal to the vector field V. Thus, our theorem links the existence of an object that belongs to some level of an extension tower with the existence of objects defined by means of the base differential ring k[x,y,z]. A self-contained proof of this result is given in the language of differential algebra. This method of finding first 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.