On maps of CR manifolds and transformations of differential equations (Q2756226)

Let \(M\) be a real analytic (Levi nondegenerate) hypersurface in \(\mathbb{C}^2\), defined locally by \(r(Z,\overline Z)=2\text{Re} w +|z|^2 +o(|Z|^2)=0\) where \(Z=(z,w)\). For any point \(p\) in a neighborhood of the origin of \(\mathbb{C}^2\) the Segre variety \(Q(p)\) is a complex curve defined by \(\{Z: r(Z,\overline p)=0\}\). Such a curve is the graph of a holomorphic function \(w(z)\) near the origin.NEWLINENEWLINENEWLINEThe author observes that \(w=w(z)\) satisfies a second order ODE \(w''=F(z,w,w')\) and the 2-parametric family of Segre varieties (the Segre family) is formed by the graphs of solutions. Then the author establishes a correspondence between CR maps of such \(M\)'s and transformations of associated ODE's defining their respective Segre families. The author proves a result on the finite determinacy and parametrization of such transformations for a wide class of such holomorphic PDE systems.