Liouvillian integrability of three dimensional vector fields

The focus of this paper is on three dimensional complex polynomial (or rational) vector fields that admit a Liouvillian first integral. We include preliminary results about one-forms over complex rational function fields, proving, resp.\ reproving an explicit characterization of closed one-forms, and Singer's theorem about Liouvillian integrability. Subsequently we consider a three dimensional complex polynomial vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. For these forms we prove that there exists a first integral whose differential is the product of a rational 1-form with a Darboux function, or there exists a Darbouxian Jacobi multiplier. (In the former case, the search for Liouvillian first integrals is reduced to the search for invariant algebraic surfaces and their associated exponential factors.) Moreover, we prove that Liouvillian integrability always implies the existence of a first integral that is -- in a well-defined sense -- defined over a finite algebraic extension of the rational function field.