Real ellipsoids and double valued reflection in complex space

We develop the theory of double valued reflection for a real ellipsoidal hypersurface M in complex Euclidean n-space. This leads to a pair of meromorphic involutions on the complexification of M. The main problem is to understand the dynamics of their composition, which is a reversible contact trnasformation. By means of the Segre polar correspondence, This map is transformed into a kind of generalized complex "null-cone billiard" map relative to two nonsingular n-dimensional complex quadrics in projective (n + 1)-space. Next we develop a generalization of classical confocal ellipsoidal coordinates, which we use, together with the Maurer-Cartan equations of the orthogonal group, to demonstrate the integrable nature of the dynamics of the reversible map. We show that certain complex integral curves of the characteristic systems of suitably invariant contact forms of the structure are invariant by the map. The integration of these systems is, in turn, reduced to solving systems of Abelian differential equations by means of generalized Jacobi inversion. Finally, we indicate how to choose the contact form so that the real loci of these characteristic curves will be closed curves. The complex curves will then yield generalized stationary curves, in the sense of Lempert, for the ellipsoidal domain.