On transformations of analytic CR-structures

We establish a link between the CR-geometry of real-analytic submanifolds in $\mathbb C^n$ and the geometry of differential equations. The idea of our approach is to regard biholomorphisms of a Levi-non-degenerate real-analytic CR-manifold $\mathscr M$ as point Lie symmetries of the second-order holomorphic system of differential equations defining the Segre family of $\mathscr M$. This enables us to study the biholomorphism group of $\mathscr M$ by means of the geometric theory of differential equations. We give several examples and applications to CR-geometry: results on the finite-dimensionality of the biholomorphism group and precise estimates of its dimension, and an explicit parametrization of the Lie algebra of infinitesimal automorphisms.