Extensions of differential flat fields and Liouvillian systems

In this paper, we introduce Liouvillian systems as a natural extension of differentially flat systems. As for flat systems, Liouvillian systems emphasize the central role played by the notion of trajectory. The study of Liouvillian systems is placed within the framework of differential algebra and of differential Galois theory. A necessary and sufficient condition for a nonflat system to be Liouvillian is given.