On nonlinear differential Galois theory (Q696192)

The author continues his paper [see Le groupoide de Galois d'un feuilletage, Monogr. Enseign. Math. 38, 465-501 (2001; Zbl 1033.32020)] discussing a new Galois theory of nonlinear differential equations. Let \(X\) denote a (smooth) complex analytic manifold, and let \(\Aut(X)\) be the space of germs of invertible maps \((X,a)\to (X,b)\), \(a,b\in X\). Some subgroupoids of \(\Aut (X)\) (Lie groupoids) defined by a system of partial differential equations are discussed in the paper. The author attaches such a groupoid to a foliation with singularities on \(X\) and calls it ``the Galois groupoid of the foliation''. In this paper he considers several new examples for the theory. Other approaches for building nonlinear differential Galois theory were provided by \textit{H. Umemura} [algebraic case, see Nagoya Math. J. 144, 59-135 (1996; Zbl 0878.12002)] and \textit{J. F. Pommaret} [analytic case, see Differential Galois theory, New York, Gordon \& Bread (1983; Zbl 0539.12013)].