Equivalences of PDE systems associated to degenerate para-CR structures: foundational aspects

Let $${\mathbb {K}}= {\mathbb {R}}$$ K = R or $${\mathbb {C}}$$ C . We study basic invariants of submanifolds of solutions $${\mathscr {M}} = \{ y = Q(x,a,b)\} = \{b = P(a,x,y)\}$$ M = { y = Q ( x , a , b ) } = { b = P ( a , x , y ) } in coordinates $$x \in {\mathbb {K}}^{n\geqslant 1}$$ x ∈ K n ⩾ 1 , $$y \in {\mathbb {K}}$$ y ∈ K , $$a \in {\mathbb {K}}^{m\geqslant 1}$$ a ∈ K m ⩾ 1 , $$b \in {\mathbb {K}}$$ b ∈ K under split-diffeomorphisms $$(x,y,a,b) \,\longmapsto \, \big ( f(x,y),\,g(x,y),\,\varphi (a,b),\,\psi (a,b) \big )$$ ( x , y , a , b ) ⟼ ( f ( x , y ) , g ( x , y ) , φ ( a , b ) , ψ ( a , b ) ) . Two Levi forms exist, and have the same rank $$r \leqslant \mathsf{min}(n,m)$$ r ⩽ min ( n , m ) . If $${\mathscr {M}}$$ M is k-nondegenerate with respect to parameters and l-nondegenerate with respect to variables, $$\mathsf{Aut}({\mathscr {M}})$$ Aut ( M ) is a local Lie group of dimension: $$\begin{aligned} \mathsf{dim}\, \mathsf{Aut}({\mathscr {M}}) \leqslant (n+1)\, \genfrac(){0.0pt}1{n+1+2k+2l}{n+1} + (m+1)\, \genfrac(){0.0pt}1{m+1+2k+2l}{m+1}. \end{aligned}$$ dim Aut ( M ) ⩽ ( n + 1 ) n + 1 + 2 k + 2 l n + 1 + ( m + 1 ) m + 1 + 2 k + 2 l m + 1 . Mainly, our goal is to set up foundational material addressed to CR geometers. We focus on $$n = m = 2$$ n = m = 2 , assuming $$r = 1$$ r = 1 . In coordinates (x, y, z, a, b, c), a local equation is: $$\begin{aligned} z = c + xa + \beta \,xxb + {\underline{\beta }}\,yaa + c\,\mathrm{O}_{x,y,a,b}(2) + \mathrm{O}_{x,y,a,b,c}(4), \end{aligned}$$ z = c + x a + β x x b + β ̲ y a a + c O x , y , a , b ( 2 ) + O x , y , a , b , c ( 4 ) , with $$\beta $$ β and $${\underline{\beta }}$$ β ̲ representing the two 2-nondegeneracy invariants at 0. The associated para-CR pde system: $$\begin{aligned} z_y = F\big (x,y,z,z_x,z_{xx}\big ) \quad and \quad z_{xxx} = H\big (x,y,z,z_x,z_{xx}\big ), \end{aligned}$$ z y = F ( x , y , z , z x , z xx ) a n d z xxx = H ( x , y , z , z x , z xx ) , satisfies $$F_{z_{xx}} \equiv 0$$ F z xx ≡ 0 from Levi degeneracy. We show in details that the hypothesis of 2-nondegeneracy with respect to variables is equivalent to $$F_{z_x z_x} \ne 0$$ F z x z x ≠ 0 . This gives a CR-geometric meaning to the first two para-CR relative differential invariants encountered independently in another paper, joint with Paweł Nurowski.