  Let $K = R$ or $C$. We study basic invariants of submanifolds of solutions
$\mathcal{M} = \{ y = Q(x,a,b)\} = \{b = P(a,x,y)\}$ in coordinates $x \in
K^{n\geqslant 1}$, $y \in K$, $a \in K^{m\geqslant 1}$, $b \in K$ under
split-diffeomorphisms $(x,y,a,b) \,\longmapsto\, \big(
f(x,y),\,g(x,y),\,\varphi(a,b),\,\psi(a,b) \big)$. Two Levi forms exist, and
have the same rank $r \leqslant \min (n,m)$. If $\mathcal{M}$ is
$k$-nondegenerate with respect to parameters and $l$-nondegenerate with respect
to variables, $\mbox{Aut}(\mathcal{M})$ is a local Lie group of dimension: \[
\dim\, \mbox{Aut} (\mathcal{M}) \,\,\leqslant\,\,
{\textstyle{\binom{n+1+2k+2l}{2k+2l}}}\,\, \min\, \big\{ (n+1),\, (m+1) \big\}.
\] Mainly, our goal is to set up foundational material addressed to CR
geometers. We focus on $n = m = 2$, assuming $r = 1$. In coordinates $(x,y,z,
a,b,c)$, a local equation is: \[ z \,=\, c + xa + \beta\,xxb +
\underline{\beta}\,yaa + c\,{\rm O}_{x,y,a,b}(2) + {\rm O}_{x,y,a,b,c}(4), \]
with $\beta$ and $\underline{\beta}$ representing the two $2$-nondegeneracy
invariants at $0$. The associated para-CR PDE system: \[ z_y \,=\,
F\big(x,y,z,z_x,z_{xx}\big) \ \ \ \ \ \ \ \ \ \ \ \ \ \& \ \ \ \ \ \ \ \ \ \ \
\ \ z_{xxx} \,=\, H\big(x,y,z,z_x,z_{xx}\big), \] satisfies $F_{z_{xx}} \equiv
0$ from Levi degeneracy.
  We show in details that the hypothesis of $2$-nondegeneracy with respect to
variables is equivalent to $F_{z_x z_x} \neq 0$. This gives CR-geometric
meaning to the first two para-CR relative differential invariants encountered
independently in arxiv.org/abs/2003.08166/.
