  To any completely integrable second-order system of real or complex partial
differential equations in n > 1 independent variables and in one dependent
variable, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan)
connection, and he computed its curvature. By means of a natural transfer of
jet polynomials to the associated submanifold of solutions, what the vanishing
of the Hachtroudi curvature gives can be precisely translated in order to
characterize when both families of Segre varieties and of conjugate Segre
varieties associated to a Levi nondegenerate real analytic hypersurface M in
C^{n+1} can be straightened to be affine complex (conjugate) hyperplanes. In
continuation to a previous paper devoted to the quite distinct C^2-case, this
then characterizes in an effective way those hypersurfaces of C^{n+1} in higher
complex dimension n+1 > 2 that are locally biholomorphic to a piece of the
(2n+1)-dimensional Heisenberg sphere, without any special assumption on their
defining equations.
