On degenerate para-CR structures: Cartan reduction and homogeneous models

Abstract

Motivated by recent works in Levi degenerate CR geometry, this article endeavours to study the wider and more flexible para-CR structures for which the constraint of invariancy under complex conjugation is relaxed. We consider $5$-dimensional para-CR structures whose Levi forms are of constant rank $1$ and that are $2$-nondegenerate both with respect to parameters and to variables. Eliminating parameters, such structures may be represented modulo point transformations by pairs of PDEs $z_y=F(x,y,z,z_x)$ $\&$ $z_{xxx}=H(x,y,z,z_x,z_{xx})$, with $F$ independent of $z_{xx}$ and $F_{z_x{z}_x}\ne 0$, that are completely integrable $D_x^{3}F={\Delta }_{y}H$, Performing at an advanced level Cartan's method of equivalence, we determine all concerned homogeneous models, together with their symmetries: (i) $z_y=\frac{1}{4}(z_x{)}^2\&z_{xxx}=0$; (ii) $z_y=\frac{1}{4}(z_x{)}^2\&z_{xxx}=(z_{xx}{)}^3$; (iiia) $z_y=\frac{1}{4}(z_x{)}^b\&z_{xxx}=(2-b)\frac{(z_{xx}{)}^2}{z_x}$ with $z_x>0$ for any real $b\in [1,2)$; (iiib) $z_y=f(z_x)\&z_{xxx}=h(z_x)(z_{xx}{)}^2$, where the function $f$ is determined by the implicit equation: \[ (z_x^2+f(z_x)^2)\, \mathrm{exp} \left( 2b\,\mathrm{arctan}\tfrac{bz_x-f(z_x)}{z_x+bf(z_x)} \right) = 1+b^2 \] and where: \[ h(z_x) := \frac{(b^2-3)z_x-4bf(z_x)}{(f(z_x)-bz_x)^2}, \] for any real $b>0$.

37 pages, 1 figure, intensive symbolic computations