On Degenerate Para-CR Structures: Cartan Reduction and Homogeneous Models

Abstract

Motivated by recent works in Levi degenerate CR geometry, this article endeavors 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 $F_{z_{x}z_{x}} \neq 0$, 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) $z_{y} = \tfrac 14 (z_{x})^{2}\quad \&\quad z_{xxx}=0$;

• (ii) $z_{y} = \tfrac 14 (z_{x})^{2}\quad \& \quad z_{xxx}=(z_{xx})^{3}$;

• (iiia) $z_{y} = \tfrac 14 (z_{x})^{b} \& z_{xxx} = (2-b)\frac {(z_{xx})^{2}}{z_{x}}$ with z_x > 0 for any real b ∈ [1,2);

• (iiib) $z_{y} = f(z_{x})\quad \& \quad z_{xxx} = h(z_{x})\left (z_{xx}\right )^{2}$, where the function f is determined by the implicit equation:

  $$ \left( {z_{x}^{2}}+f(z_{x})^{2}\right) \exp \left( 2b \text{arctan}\tfrac{bz_{x}-f(z_{x})}{z_{x}+bf(z_{x})} \right) = 1+b^{2} $$

  and where, for any real b > 0:

  $$ h(z_{x}) := \frac{(b^{2}-3)z_{x}-4bf(z_{x})}{\left( f(z_{x})-bz_{x}\right)^{2}}. $$