Segre varieties , CR geometry and Lie symmetries of second order PDE systems

We establish a link between the CR geometry of real analytic submanifolds in I C and the geometric PDE theory. The main idea of our approach is to consider biholomorphisms of a Levi-nondegenerate real analytic Cauchy-Riemann manifold M as poinwise symmetries of a second order holomorphic PDE system defining the Segre family of M. This allows to employ the well-elaborated PDE tools in order to study the biholomorphism group of M. We give several examples and applications to the CR geometry: the results on the finite dimensionality of the biholomorphism group and precise estimates of its dimension, explicit parametrization of the Lie algebra of infinitesimal automorphisms etc. We deduce these results as a special case of more general statements concerning related properties of symmetries of second order PDE systems. AMS Mathematics Subject Classification: 32H, 32M.