$\displaystyle\hphantom{\big{[}\tilde{X}_{i},\tilde{X}_{l}\big{]}}{}=\frac{1}{X_{i}(K)X_{l}(I)}\left([X_{i},X_{l}]-B_{il}^{l}X_{l}-B_{il}^{i}X_{i}-B_{il}^{j}X_{j}\right)=0.$

An analogous calculation using
$B_{jl}^{j}X_{j}(K)+B_{jl}^{i}X_{i}(K)=-X_{l}X_{j}(K)$ and
$B_{jl}^{l}X_{l}(I)=X_{j}X_{l}(I)$ shows that
$\big{[}\tilde{X}_{j},\tilde{X}_{l}\big{]}=0$. ∎

The formulation of the Darboux-adapted coframe, as well as the construction of
conservation laws for Darboux integrable systems, will rely on our ability to
obtain characteristic invariant contact forms from the invariant functions
whose existence is guaranteed in Lemma 4.7. For example, we can write down a
contact form which is invariant with respect to the characteristic vector
fields $X_{1}$ and $X_{2}$ as follows,

######  Lemma 4.10.

Let $I$, $J$, and $K$ be functions on $\mathcal{R}^{\infty}$ that are
invariant with respect to the characteristic vector field $X_{3}$ such that
$X_{1}(I)=1$, $X_{2}(I)=0$, $X_{1}(J)=0$, and $X_{2}(J)=1$. Also define
$K^{\prime}=X_{1}(K)$ and $K^{\prime\prime}=X_{2}(K)$. Then

$\displaystyle\omega={\rm d}_{V}K-K^{\prime}{\rm d}_{V}I-K^{\prime\prime}{\rm
d}_{V}J$

is an $X_{3}$-invariant contact form.

###### Proof.

Note that $K^{\prime}$ and $K^{\prime\prime}$ are also $X_{3}$ invariant since
the characteristic vector fields $X_{i}$ all commute. Computing ${\rm
d}_{H}\omega$, we obtain

$\displaystyle{\rm d}_{H}\omega=-{\rm d}_{V}{\rm d}_{H}K-{\rm
d}_{H}K^{\prime}\wedge{\rm d}_{V}I+K^{\prime}{\rm d}_{V}{\rm d}_{H}I-{\rm
d}_{H}K^{\prime\prime}\wedge{\rm d}_{V}J+K^{\prime\prime}{\rm d}_{V}{\rm
d}_{H}J$ $\displaystyle\hphantom{{\rm d}_{H}\omega}{}=-{\rm
d}_{V}(K^{\prime}\sigma_{1}+K^{\prime\prime}\sigma_{2})-(X_{1}(K^{\prime})\sigma_{1}+X_{2}(K^{\prime})\sigma_{2})\wedge{\rm
d}_{V}I$ $\displaystyle\hphantom{{\rm d}_{H}\omega=}{}+K^{\prime}{\rm
d}_{V}(\sigma_{1})-(X_{1}(K^{\prime\prime})\sigma_{1}+X_{2}(K^{\prime\prime})\sigma_{2})\wedge{\rm
d}_{V}J+K^{\prime\prime}{\rm d}_{V}(\sigma_{2})$ $\displaystyle\hphantom{{\rm
d}_{H}\omega}{}=\sigma_{1}\wedge{\rm d}_{V}K^{\prime}+\sigma_{2}\wedge{\rm
d}_{V}K^{\prime\prime}-X_{1}(K^{\prime})\sigma_{1}\wedge{\rm
d}_{V}I-X_{2}(K^{\prime})\sigma_{2}\wedge{\rm d}_{V}I$
$\displaystyle\hphantom{{\rm
d}_{H}\omega=}{}-X_{1}(K^{\prime\prime})\sigma_{1}\wedge{\rm
d}_{V}J-X_{2}(K^{\prime\prime})\sigma_{2}\wedge{\rm d}_{V}J$
$\displaystyle\hphantom{{\rm d}_{H}\omega}{}=\sigma_{1}\wedge({\rm
d}_{V}K^{\prime}-X_{1}(K^{\prime}){\rm d}_{V}I-X_{1}(K^{\prime\prime}){\rm
d}_{V}J)$ $\displaystyle\hphantom{{\rm d}_{H}\omega=}{}+\sigma_{2}\wedge({\rm
d}_{V}K^{\prime\prime}-X_{2}(K^{\prime}){\rm
d}_{V}I-X_{2}(K^{\prime\prime}){\rm d}_{V}J).$

From this calculation we see that $X_{3}(\omega)=X_{3}\,\lrcorner\,{\rm
d}_{H}(\omega)=0$ so that $\omega$ is $X_{3}$-invariant. ∎

Likewise,we can use $X_{1}$ and $X_{2}$ invariant functions to obtain $X_{1}$
and $X_{2}$ invariant contact forms in the following way.

######  Lemma 4.11.

Let $I$ and $J$ be functions on $\mathcal{R}^{\infty}$ which are invariant
with respect to both characteristic vector fields $X_{1}$ and $X_{2}$, such
that $X_{3}(I)=I^{\prime}$ and $X_{3}(J)=1$. Then

$\displaystyle\omega={\rm d}_{V}I-I^{\prime}{\rm d}_{V}J$

is an $X_{1}$ and $X_{2}$ invariant contact form.

###### Proof.

In the same spirit as the last proof, we compute ${\rm d}_{H}\omega$ and by
grouping terms arrive at

$\displaystyle{\rm d}_{H}\omega=\sigma_{3}\wedge({\rm
d}_{V}I^{\prime}-X_{3}(I^{\prime}){\rm d}_{V}J).$

So $X_{1}\,\lrcorner\,{\rm d}_{H}(\omega)=X_{2}\,\lrcorner\,{\rm
d}_{H}(\omega)=0$ and $\omega$ is both $X_{1}$ and $X_{2}$ invariant. ∎

Given the preceding lemma, the following corollary follows immediately from
Proposition 3.14.

######  Corollary 4.12.

Let $\mathcal{R}$ be a system of equations of the form (4.12). If
$\mathcal{R}$ satisfies the conditions of Lemma 4.7, then at least one of the
Laplace indices $p_{ij}=\operatorname{ind}(\mathcal{X}_{ij})$ must be finite.

The construction of the Darboux adapted coframe will proceed according to the
orders of the various characteristic invariant functions in Lemma 4.7. Here we
do not have a complete analysis of the possible orders of invariants, as was
given by Goursat [19] in the case of a single hyperbolic second order equation
in the plane. Thus we will carry out the construction under certain
assumptions on the orders of the invariants to serve as a demonstration of how
a coframe may be constructed in a particular case.

Given two first order $X_{1}$ and $X_{2}$ invariant functions, $I_{1}$ and
$\tilde{I}_{1}$, we may obtain a sequence of functions invariant with respect
to $X_{1}$ and $X_{2}$ by repeatedly applying $X_{3}$:

$\displaystyle\tilde{I}_{1},I_{1},I_{2}=X_{3}(I_{1}),I_{3}=X_{3}(I_{2}),\ldots,\qquad\textrm{where}\quad
X_{3}\big{(}\tilde{I}_{1}\big{)}=1.$

Then by Lemma 4.11, the following are a sequence of $X_{1}$ and $X_{2}$
invariant contact forms,

$\displaystyle\alpha_{1}={\rm d}_{V}I_{1}-I_{2}{\rm
d}_{V}\tilde{I}_{1},\quad\alpha_{2}={\rm d}_{V}I_{2}-I_{3}{\rm
d}_{V}\tilde{I}_{1},\quad\ldots,\quad\alpha_{i}={\rm d}_{V}I_{i}-I_{i+1}{\rm
d}_{V}\tilde{I}_{1},\quad\dots.$

For $i\geq 2$, equation (4.11) implies that

$\displaystyle\alpha_{i}\equiv
a_{i}\hat{\xi}_{3}^{i}\mod\big{\\{}\theta,\hat{\xi}_{1}^{1},\ldots,\hat{\xi}_{1}^{i-1},\xi_{2}^{1},\ldots,\hat{\xi}_{2}^{i-1}\big{\\}},$

where $a_{i}\neq 0$. For $i=1$, we have $\alpha_{1}\equiv
a_{1}\hat{\xi}_{3}^{1}\mod\\{\theta\\}$ with $a_{1}\neq 0$, as we will now
show. Since ${\rm
d}_{H}\theta\equiv\sigma_{1}\wedge\hat{\xi}_{1}^{1}+\sigma_{2}\wedge\hat{\xi}_{2}^{1}+\sigma_{3}\wedge\hat{\xi}_{3}^{1}\mod\\{\theta\\}$,
if $a_{1}=0$ then

$\displaystyle{\rm d}_{H}\alpha_{1}\equiv
0\quad\mod\big{\\{}\theta,\hat{\xi}_{1}^{1},\hat{\xi}_{2}^{1},\hat{\xi}_{3}^{1}\big{\\}}.$
(4.22)

But since ${\rm d}_{H}\alpha_{1}=\sigma_{3}\wedge\alpha_{2}\equiv
a_{2}\sigma_{3}\wedge\hat{\xi}_{3}^{2}\mod\big{\\{}\theta,\hat{\xi}_{1}^{1},\hat{\xi}_{2}^{1},\hat{\xi}_{3}^{1}\big{\\}}$,
equation (4.22) contradicts the fact that $\alpha_{2}$ is of order 2, so
indeed $a_{1}\neq 0$.

The set of contact forms $\\{\alpha_{1},\alpha_{2},\ldots\\}$ can thus replace
the branch of the Laplace coframe given by
$\big{\\{}\hat{\xi}_{3}^{1},\hat{\xi}_{3}^{2},\ldots\big{\\}}$ previously. The
structure equations for the $\alpha_{i}$ are given by

$\displaystyle{\rm d}\alpha_{i}={\rm d}\tilde{I}_{1}\wedge\alpha_{i+1},$
(4.23)

where (4.23) is calculated in the following way

$\displaystyle{\rm d}\alpha_{i}={\rm d}({\rm d}_{V}I_{i})-{\rm
d}(I_{i+1})\wedge{\rm d}_{V}\tilde{I}_{1}-I_{i+1}{\rm d}({\rm
d}_{V}\tilde{I}_{1})={\rm d}_{H}{\rm d}_{V}I_{i}-{\rm d}(I_{i+1})\wedge{\rm
d}_{V}\tilde{I}_{1}-I_{i+1}{\rm d}_{H}{\rm d}_{V}\tilde{I}_{1}$
$\displaystyle\hphantom{{\rm d}\alpha_{i}}{}=-{\rm
d}_{V}(I_{i+1}\sigma_{3})-(I_{i+2}\sigma_{3}+{\rm d}_{V}I_{i+1})\wedge{\rm
d}_{V}\tilde{I}_{1}+I_{i+1}{\rm d}_{V}(\sigma_{3})$
$\displaystyle\hphantom{{\rm d}\alpha_{i}}{}=-{\rm
d}_{V}I_{i+1}\wedge\sigma_{3}-(I_{i+2}\sigma_{3}+{\rm d}_{V}I_{i+1})\wedge{\rm
d}_{V}\tilde{I}_{1}$ $\displaystyle\hphantom{{\rm
d}\alpha_{i}}{}=(\sigma_{3}+{\rm d}_{V}\tilde{I}_{1})\wedge({\rm
d}_{V}I_{i+1}-I_{i+2}{\rm d}_{V}\tilde{I}_{1})={\rm
d}\tilde{I}_{1}\wedge\alpha_{i+1}.$

To complete the coframe, we need to construct contact forms which are
equivalent to $\big{\\{}\hat{\xi}_{1}^{1},\hat{\xi}_{1}^{2},\ldots\big{\\}}$
and $\big{\\{}\hat{\xi}_{2}^{1},\hat{\xi}_{2}^{2},\ldots\big{\\}}$. We
accomplish this by first generating $X_{3}$ invariant functions, repeatedly
applying the total vector fields $X_{1}$ and $X_{2}$ to the $X_{3}$ invariant
functions $J$ and $K$ from Lemma 4.7 to obtain the following two sequences of
$X_{3}$ invariant functions:

$\displaystyle\tilde{J},J=J_{1},J_{2}=X_{1}(J_{1}),J_{3}=X_{1}(J_{2}),\ldots,$
$\displaystyle\tilde{K},K=K_{1},K_{2}=X_{2}(K_{1}),K_{3}=X_{2}(K_{2}),\ldots.$

Then using Lemma 4.10, we can write two corresponding sequences of $X_{3}$
invariant contact forms:

$\displaystyle\beta_{1}={\rm d}_{V}J_{1}-X_{1}(J_{1}){\rm
d}_{V}\tilde{K}-X_{2}(J_{1}){\rm d}_{V}\tilde{J},$
$\displaystyle\beta_{2}={\rm d}_{V}J_{2}-X_{1}(J_{2}){\rm
d}_{V}\tilde{K}-X_{2}(J_{2}){\rm d}_{V}\tilde{J},$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle\beta_{i}={\rm d}_{V}J_{i}-X_{1}(J_{i}){\rm
d}_{V}\tilde{K}-X_{2}(J_{i}){\rm d}_{V}\tilde{J}$

and

$\displaystyle\gamma_{1}={\rm d}_{V}K_{1}-X_{1}(K_{1}){\rm
d}_{V}\tilde{K}-X_{2}(K_{1}){\rm d}_{V}\tilde{J},$
$\displaystyle\gamma_{2}={\rm d}_{V}K_{2}-X_{1}(K_{2}){\rm
d}_{V}\tilde{K}-X_{2}(K_{2}){\rm d}_{V}\tilde{J},$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle\gamma_{i}={\rm d}_{V}K_{i}-X_{1}(K_{i}){\rm
d}_{V}\tilde{K}-X_{2}(K_{i}){\rm d}_{V}\tilde{J}.$

Then a similar argument to that given above for the sequence of $\alpha_{i}$
shows that

$\displaystyle\beta_{i}\equiv
b_{k}\hat{\xi}_{1}^{i}+c_{k}\hat{\xi}_{2}^{i}\quad\mod\big{\\{}\theta,\hat{\xi}_{3}^{1},\hat{\xi}_{3}^{2},\ldots,\hat{\xi}_{3}^{i-1}\big{\\}}$

and likewise,

$\displaystyle\gamma_{i}\equiv
e_{k}\hat{\xi}_{1}^{i}+f_{k}\hat{\xi}_{2}^{i}\quad\mod\big{\\{}\theta,\hat{\xi}_{3}^{1},\hat{\xi}_{3}^{2},\ldots,\hat{\xi}_{3}^{i-1}\big{\\}},$

where at least one of $b_{k}$ and $c_{k}$ is nonzero, and at least one of
$e_{k}$ and $f_{k}$ is nonzero. While Lemma 4.7 requires that ${\rm
d}J\wedge{\rm d}\tilde{J}\wedge{\rm d}K\wedge{\rm d}\tilde{K}\neq 0$, this
condition is not enough to guarantee that ${\rm d}_{V}J_{i}$ and ${\rm
d}_{V}K_{i}$ are functionally independent. As a result, there is some
ambiguity about the construction of the remaining two branches of the coframe,
as it will depend on whether the coefficients on $\hat{\xi}_{1}^{i}$ and
$\hat{\xi}_{2}^{i}$ in $\beta_{i}$ and $\gamma_{i}$ satisfy
$b_{k}f_{k}-c_{k}e_{k}\neq 0$. In other words on whether $\beta_{i}$ and
$\gamma_{i}$ are independent ${\rm
mod}\big{\\{}\theta,\hat{\xi}_{3}^{1},\hat{\xi}_{3}^{2},\ldots,\hat{\xi}_{3}^{i-1}\big{\\}}$.
If any pair $\beta_{i}$ and $\gamma_{i}$ does fail to be independent, then we
will complete the coframe by taking the necessary contact forms from the
Laplace adapted coframe.

This coframe would likely be useful in the pursuit of an eventual
classification of Darboux-integrable systems, similar to that presented in
[20] for the case of a scalar second order hyperbolic PDE in the plane.

###  4.3 Generating infinitely many $\boldsymbol{(1,s)}$ and
$\boldsymbol{(2,s)}$ conservation laws

We may assume that for a Darboux integrable system,
$[X_{i},X_{l}]=P_{il}^{i}X_{i}+P_{il}^{l}X_{l}=0$ and
$[X_{j},X_{l}]=P_{jl}^{j}X_{j}+P_{jl}^{l}X_{l}=0$. Recalling the structure
equation

$\displaystyle{\rm
d}_{H}\sigma_{l}=P_{il}^{l}\sigma_{i}\wedge\sigma_{l}+P_{jl}^{l}\sigma_{j}\wedge\sigma_{l}$

we see that if $X_{l}$ commutes with $X_{i}$ and $X_{j}$, then ${\rm
d}_{H}\sigma_{l}=0$. With this in mind, we may now demonstrate a way to
construct infinitely many $(1,s)$ conservation laws for Darboux integrable
systems.

######  Theorem 4.13.

Let $\mathcal{R}$ be a Darboux integrable system of equations. Then there
exist infinitely many non-trivial type $(1,s)$ conservation laws for all
$s\geq 0$.

###### Proof.

Fix distinct $i$, $j$, and $l$ as in Lemma 4.7. For $s=0$, we may construct a
conservation law by taking $\omega=I\sigma_{l}$ where $I$ is an $X_{i}$ and
$X_{j}$ invariant function. Suppose that $\omega$ is exact, that is
$\omega=I\sigma_{l}={\rm d}_{H}\tilde{I}$ for some function $\tilde{I}$. Then

$\displaystyle{\rm
d}_{H}\tilde{I}=X_{1}(\tilde{I})\sigma_{1}+X_{2}(\tilde{I})\sigma_{2}+X_{3}(\tilde{I})\sigma_{3}=I\sigma_{l},$

so $\tilde{I}$ is $X_{i}$ and $X_{j}$ invariant and $X_{l}(\tilde{I})=I$. Thus
$\omega$ will be a nontrivial conservation law so long as $I\neq
X_{l}(\tilde{I})$ for $\tilde{I}$ another $X_{i}$ and $X_{j}$ invariant
function. Additional $(1,0)$ conservation laws may be constructed by
generating additional $X_{i}$ and $X_{j}$ invariant functions by applying
$X_{l}$ to $I$ repeatedly. Note that $X_{l}(I)$ is still $X_{i}$ and $X_{j}$
invariant since $X_{l}$ commutes with both $X_{i}$ and $X_{j}$.

For $s\geq 1$, we first note that if $\omega$ is a $d$ closed $(s+1)$ form
with vanishing $(2,s-1)$ component, then the $(1,s)$ component of $\omega$
will be ${\rm d}_{H}$ closed. To see this, write ${\rm d}\omega$ as a sum of
forms of different bidegrees, according to the direct sum decomposition of the
space of $(s+2)$ forms,

$\displaystyle\Omega^{s+2}(\mathcal{R}^{\infty})=\bigoplus_{0\leq i\leq
3}\Omega^{(i,s+2-i)}(\mathcal{R}^{\infty}).$ (4.24)

Let ${\rm d}\omega=\nu_{0}+\nu_{1}+\nu_{2}+\nu_{3}$ where
$\nu_{i}\in\Omega^{(i,s+2-i)}$. Since each $\nu_{i}$ belongs to a different
component of (4.24), ${\rm d}\omega=0$ implies that $\nu_{i}=0$ for all $i$.
In particular, $\nu_{2}=0$. The form $\omega$ can itself be decomposed
according to bidegree as well. Write $\omega=\mu_{0}+\mu_{1}+\mu_{2}+\mu_{3}$
where $\mu_{i}\in\Omega^{(i,s+1-i)}$. Now decompose ${\rm d}\omega$ into its
horizontal and vertical components, ${\rm d}\omega=({\rm d}_{H}+{\rm
d}_{V})(\omega)$, to see that ${\rm d}_{H}\mu_{1}+{\rm
d}_{V}\mu_{2}=\nu_{2}=0$. If the $(2,s-1)$ component, $\mu_{2}$, of $\omega$
vanishes, then this last equation becomes ${\rm d}_{H}\mu_{1}=0$ and we can
conclude that the $(1,s)$ component of $\omega$ is ${\rm d}_{H}$ closed.

Now, to construct a ${\rm d}_{H}$ closed $(1,s)$ form, we begin by defining
$\alpha={\rm d}I_{1}\wedge{\rm d}I_{2}\wedge\cdots\wedge{\rm d}I_{s+1}$ where
each of the functions $I_{n}$ is an $X_{i}$ and $X_{j}$ invariant function.
These functions can be obtained as before by taking $I=I_{1}$ and repeatedly
applying the total vector field $X_{l}$ to $I$ to generate the remaining
functions. The $(s+1)$ form $\alpha$ will have vanishing $(2,s-1)$ component:
since the functions $I_{n}$ are all $X_{i}$ and $X_{j}$ invariant, the only
horizontal form that will appear in any of the ${\rm d}I_{n}$ is $\sigma_{l}$.
Thus, by the preceding discussion,

$\displaystyle\pi^{1,s}({\rm d}I_{1}\wedge{\rm d}I_{2}\wedge\cdots\wedge{\rm
d}I_{s+1})$

will be ${\rm d}_{H}$ closed as required.

Furthermore, we can construct a ${\rm d}_{H}$ closed $(1,s)$ form that is not
exact. In other words, we can write down a nontrivial $(1,s)$ conservation
law. Let
$p_{l}=\min\\{\operatorname{ind}(\mathcal{X}_{il}),\operatorname{ind}(\mathcal{X}_{jl})\\}$,
and take $k\geq p_{l}+s$. Then for a nonzero form
$\eta\in\Omega_{\mathcal{I}}^{s-2}(\alpha_{p_{l}+1},\alpha_{p_{l}+2},\ldots,\alpha_{k-1})$
where $\mathcal{I}$ is the ring of functions which are both $X_{i}$ and
$X_{j}$ invariant, the $(1,s)$ form

$\displaystyle\omega=\sigma_{l}\wedge\alpha_{k+1}\wedge\alpha_{k}\wedge\eta$
(4.25)

is ${\rm d}_{H}$ closed, but not ${\rm d}_{H}$ exact. It is straightforward to
see that $\omega$ is indeed ${\rm d}_{H}$ closed since $\sigma_{l}$ is itself
${\rm d}_{H}$ closed and $\alpha_{k+1}\wedge\alpha_{k}\wedge\eta$ is both
$X_{i}$ and $X_{j}$ invariant. Now suppose that $\omega$ were ${\rm d}_{H}$
exact, that is, that there exists a $(0,s)$ form, $\gamma$ such that ${\rm
d}_{H}\gamma=\omega$. Since ${\rm d}_{H}\gamma=\sigma_{1}\wedge
X_{1}(\gamma)+\sigma_{2}\wedge X_{2}(\gamma)+\sigma_{3}\wedge X_{3}(\gamma)$,
comparing this with the above expression, (4.25), for $\omega$ we conclude
that $\gamma$ must also be $X_{i}$ and $X_{j}$ invariant. Moreover,
$X_{l}(\gamma)=\alpha_{k+1}\wedge\alpha_{k}\wedge\eta$ which implies that
$\gamma\in\Omega_{\mathcal{I}}^{s}(\alpha_{p_{l}+1},\alpha_{p_{l}+2},\ldots)$.
The adapted order of $\omega$ being $k+1$ implies that the adapted order of
$\gamma$ would be $k$. Then $\gamma$ can be written as
$\gamma=\alpha_{k}\wedge\beta+\delta$ where $\beta$ and $\delta$ each have
adapted order $\leq k-1$, and are both $X_{i}$ and $X_{j}$ invariant forms.
But then
$X_{l}(\gamma)=X_{l}(\alpha_{k}\wedge\beta+\delta)=\alpha_{k+1}\wedge\alpha_{k}\wedge\eta$
which indicates that $\beta=\alpha_{k}\wedge\eta$, showing that $\beta$ has
adapted order $k$. This contradicts the original premise that $\beta$ be of
adapted order $\leq k-1$, from which we conclude that $\omega$ is not exact. ∎

It is possible to construct infinitely many nontrivial $(2,s)$ conservation
laws in a similar way. By an argument analogous to that given above in the
case of $(1,s)$ conservation laws, if $\omega$ is a ${\rm d}$ closed $(s+2)$
form with vanishing $(3,s-1)$ component, then the $(2,s)$ component of
$\omega$ is ${\rm d}_{H}$ closed. Consider the form ${\rm d}J_{1}\wedge{\rm
d}J_{2}\wedge\cdots\wedge{\rm d}J_{s+2}$ where the functions
$J_{n}=X_{i}(J_{n-1})$ are all $X_{l}$ invariant. This $(s+2)$ form has
vanishing $(3,s-1)$ component since each term ${\rm d}J_{n}$ can only contain
the horizontal forms $\sigma_{i}$ and $\sigma_{j}$. So

$\displaystyle\nu=\pi^{2,s}({\rm d}J_{1}\wedge{\rm
d}J_{2}\wedge\cdots\wedge{\rm d}J_{s+2})$

is a ${\rm d}_{H}$ closed $(2,s)$ form. Construct a second ${\rm d}_{H}$
closed $(2,s)$ form by taking the $(2,s)$ component of the $(s+2)$ form ${\rm
d}K_{1}\wedge{\rm d}K_{2}\wedge\cdots\wedge{\rm d}K_{s+2}$ where
$K_{n}=X_{j}(K_{n-1})$ and each $K_{n}$ is $X_{l}$ invariant. Let

$\displaystyle\mu=\pi^{2,s}({\rm d}K_{1}\wedge{\rm
d}K_{2}\wedge\cdots\wedge{\rm d}K_{s+2}).$

Take $\mathcal{I}$ to be the ring of $X_{l}$ invariant functions. Then the
following $(2,s)$ form will be a nontrivial conservation law

$\displaystyle\omega=\sigma_{i}\wedge\sigma_{j}\wedge[\nu_{k+1}\wedge\nu_{k}\wedge\eta_{1}+\mu_{k+1}\wedge\mu_{k}\wedge\eta_{2}],$
(4.26)

where
$\eta_{1}\in\Omega_{\mathcal{I}}^{s-2}(\nu_{p_{i}+1},\nu_{p_{i}+2},{\ldots},\nu_{k-1})$
and
$\eta_{2}\in\Omega_{\mathcal{I}}^{s-2}(\mu_{p_{j}+1},\mu_{p_{j}+2},{\ldots},\mu_{k-1})$,
and $k\geq\max\\{p_{j}+s,p_{i}+s\\}$. Clearly ${\rm d}_{H}\omega=0$ since
$\sigma_{i}\wedge\sigma_{j}$ is ${\rm d}_{H}$ closed and
$\nu_{k+1}\wedge\nu_{k}\wedge\eta_{1}$ and
$\mu_{k+1}\wedge\mu_{k}\wedge\eta_{2}$ are $X_{l}$ invariant. To see that
$\omega$ is not exact, we will assume otherwise and produce a contradiction.
Suppose that there exists a $(1,s)$ form, $\gamma$, such that ${\rm
d}_{H}\gamma=\omega$. First note that by writing ${\rm
d}_{H}\gamma=\sigma_{1}\wedge X_{1}(\gamma)+\sigma_{2}\wedge
X_{2}(\gamma)+\sigma_{3}\wedge X_{3}(\gamma)$ and looking at equation (4.26),
it may be concluded that $\gamma$ is $X_{l}$ invariant, $l\neq i,j$.
Furthermore, any $(1,s)$ form can be written as

$\displaystyle\gamma=\sigma_{1}\wedge M_{1}+\sigma_{2}\wedge
M_{2}+\sigma_{3}\wedge M_{3},$

where each $M_{i}$ is a $(0,s)$ form. Comparing equation (3.11) for ${\rm
d}_{H}\gamma$ with the expression (4.26) for $\omega$, we see that ${\rm
d}_{H}\gamma=\omega$ implies that

$\displaystyle{\rm
d}_{H}\gamma=\sigma_{i}\wedge\sigma_{j}\wedge[X_{j}(M_{i})-X_{i}(M_{j})],$

where

$\displaystyle
X_{j}(M_{i})-X_{i}(M_{j})=\nu_{k+1}\wedge\nu_{k}\wedge\eta_{1}+\mu_{k+1}\wedge\mu_{k}\wedge\eta_{2}$
(4.27)

and so
$M_{i},M_{j}\in\Omega_{\mathcal{I}}^{s}(\nu_{p_{l}+1},\nu_{p_{l}+2},\ldots)$.
Since the adapted order of $\omega$ is $(k+1)$, the adapted order of $\gamma$
must be $k$ and hence $M_{i}$ and $M_{j}$ may be written as

$\displaystyle M_{i}=\mu_{k}\wedge\alpha_{1}+\delta_{1},\qquad
M_{j}=\nu_{k}\wedge\alpha_{2}+\delta_{2},$

where $\alpha_{1}$, $\alpha_{2}$, $\delta_{1}$, and $\delta_{2}$ all have
adapted order $\leq k-1$. Plugging these expressions for $M_{i}$ and $M_{j}$
into (4.27) yields

$\displaystyle
X_{j}(\mu_{k}\wedge\alpha_{1}+\delta_{1})-X_{i}(\nu_{k}\wedge\alpha_{2}+\delta_{2})=\nu_{k+1}\wedge\nu_{k}\wedge\eta_{1}+\mu_{k+1}\wedge\mu_{k}\wedge\eta_{2}.$
(4.28)

From equation (4.28), we deduce that $\alpha_{1}=\mu_{k}\wedge\eta_{2}$ and
$\alpha_{2}=\nu_{k}\wedge\eta_{1}$. Since this contradicts the fact that
$\alpha_{1}$ and $\alpha_{2}$ were taken to have order $\leq k-1$, we can
conclude that $\omega$ is indeed not exact and thus represents a nontrivial
conservation law.

##  5 Conclusion and further research

In this paper we have proven a number of results concerning the existence and
structure of conservation laws for involutive systems of partial differential
equations of the form

$\displaystyle
u_{ij}-f_{ij}\big{(}x^{1},x^{2},x^{3},u,u_{i},u_{j}\big{)}=0\qquad\textrm{for}\quad
1\leq i<j\leq 3.$ (5.1)

After defining the constrained variational bicomplex, which provides the
framework in which we investigate form-valued conservation laws, and
developing other technical tools such as the generalized Laplace transform,
and the structure equations and Lie bracket congruences for the Laplace
adapted coframe, the first main result is proved in Section 3.5. This result,
Theorem 3.15, describes the structure of $(2,s)$ conservation laws for $s\geq
1$ in terms of solutions to the adjoint equations of the associated linearized
system. As a consequence of this structure theorem, we obtain another key
result which states that if all of the Laplace indices of the linearized
system are infinite, then all $(2,s)$ conservation laws for $s\geq 3$ are
trivial. That is, in this case the horizontal cohomology of the constrained
variational bi-complex associated to the system is trivial:

$\displaystyle H^{2,s}(\mathcal{R}^{\infty},{\rm
d}_{H})=0\qquad\textrm{for}\quad s\geq 3.$

Conditions are given in Lemma 4.7 which will guarantee that a system of the
form (1.1) will satisfy the definition of Darboux integrability introduced in
[3]. It is then show that these same conditions will imply that at least one
of the Laplace indices for the linearization of such a system must be finite.
A procedure for using characteristic invariant contact forms to construct non-
trivial $(1,s)$ and $(2,s)$ conservation laws for $s\geq 0$ is then described
in Section 4.3.

There are several natural directions in which one may try to extend the
results presented here. First, we should emphasize that in this paper we have
considered the very specific class of semi-linear systems of the form (5.1).
In particular, the characteristic vector fields, $X_{i}$, for such systems
commute. It should be fairly easy to investigate which of the results
presented here could be replicated for more general classes of nonlinear
systems belonging to the different cases of Cartan’s structural classification
listed in Theorem A.3, specifically systems with non-commuting
characteristics. Other directions in which to extend this work would be to
consider overdetermined systems in one dependent and $n$ independent
variables, instead of only 3 independent variables, or to allow $u$ to be a
vector-valued function (see, for example, [35]). Furthermore, it remains to
find further examples and applications which fall under the form of systems
(5.1) which we consider here. Constructing involutive overdetermined systems
is a project in its own right. Another interesting avenue of research would
consist in classifying Darboux-integrable systems of the form (5.1) or a
generalization of these. A potentially fruitful approach to this problem would
utilize the definitions and classification scheme presented in [3], which is
discussed in Section 4.

Beyond these points, there is the overarching question of how the horizontal
cohomology of the constrained variational bi-complex $H^{r,s}$ can be
interpreted for $s\geq 1$. In other words, what does the horizontal cohomology
mean in terms of the analytic properties of the solution space of the system
$\mathcal{R}$? Here we refer the reader to [2] for some examples of the ways
in which the cohomology of the variational bi-complex has been used to study,
for example, the inverse problem of the calculus of variations for autonomous
systems of ODE, Riemannian structures, and Gelfand–Fuks cohomology. In [28],
Tsujishita also describes several applications of the higher-degree cohomology
of the variational bi-complex, including its role in determining deformation
classes of the solution space of $\mathcal{R}$. An interpretation of
$H^{n-1,2}(\mathcal{R}^{\infty})$ cohomology, where $n$ is the number of
independent variables in the associated PDEs system, is also given in [36]. It
is shown there that the existence of nontrivial cohomology classes
$[\omega]\in H^{n-1,2}$ indicates the possible existence of variational
principles for $\mathcal{R}$. Specifically, if $\mathcal{R}$ is the
Euler–Lagrange equations for some Lagrangian $\lambda$, then it can be shown
that ${\rm d}_{V}\lambda={\rm d}_{H}\eta$ on $\mathcal{R}^{\infty}$ for $\eta$
a form of bi-degree $(n-1,1)$. Therefore $\omega={\rm d}_{V}(\eta)$, which
Zuckerman calls the universal conserved current for $\lambda$, is a ${\rm
d}_{H}$ closed $(n-1,2)$ form. Assuming that $\lambda$ is nontrivial,
Vinogradov’s two-line theorem [33] implies that $\omega$ is not ${\rm d}_{H}$
exact, so that if $H^{n-1,2}(\mathcal{R}^{\infty})=0$, then $\mathcal{R}$ does
not admit a variational principle.

##  Appendix A Structural classification

In [10], Cartan studies involutive Pfaffian systems defined on a suitable jet
space, which arise when considering involutive systems of PDEs from the
perspective of exterior differential systems. In the course of his analysis,
the Pfaffian systems’ structure equations are computed and simplified. Systems
with the same reduced structure equations are then regarded as being
structurally equivalent. Some of the main results of this work are summarized
in [25], and it is this exposition which we will follow as we proceed to place
systems of the form (1.1) in the context of Cartan’s structural
classification.

Let $\Delta^{a}(x^{i},u,u_{i},u_{ij})=0$ with $i,j=1,2,3$ be an involutive
second order system of PDEs considered as a subset of the bundle of 2-jets,
$J^{2}\big{(}\mathbb{R}^{3},\mathbb{R}\big{)}$. As defined in Section 2,
consider the contact system on $J^{2}\big{(}\mathbb{R}^{3},\mathbb{R}\big{)}$
generated by the 1-forms

$\displaystyle\theta^{0}={\rm d}u-\sum_{j=1}^{3}u_{j}{\rm
d}x^{j},\qquad\theta^{i}={\rm d}u_{i}-\sum_{j=1}^{3}u_{ij}{\rm d}x^{j},\qquad
i=1,2,3.$

Let $\omega^{i}={\rm d}x^{i}$, $\pi_{j}^{i}={\rm d}u_{ij}=\pi_{i}^{j}$ so that
a local coframe for $J^{2}\big{(}\mathbb{R}^{3},\mathbb{R}\big{)}$ is given by
$\big{\\{}\omega^{i},\theta^{0},\theta^{i},\pi_{j}^{i}\big{\\}}$ for
$i,j=1,2,3$. Then we have the following structure equations

$\displaystyle{\rm
d}\theta^{0}=\sum_{j=1}^{3}\omega^{j}\wedge\theta^{j},\qquad{\rm
d}\theta^{i}=\sum_{j=1}^{3}\omega^{j}\wedge\pi_{j}^{i}.$ (A.1)

The set of forms $\big{\\{}\theta^{0},\theta^{i}\big{\\}}$ comprises a basis
for the contact system on $J^{2}\big{(}\mathbb{R}^{3},\mathbb{R}\big{)}$.
Keeping in mind that our goal is to obtain a simplified set of structure
equations, another basis may be found by transforming the forms
$\big{\\{}\theta^{0},\theta^{i}\big{\\}}$ in the following way

$\displaystyle\left(\begin{matrix}\bar{\theta}^{0}\\\ \bar{\theta}^{1}\\\
\bar{\theta}^{2}\\\
\bar{\theta}^{3}\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&a_{1}^{1}&a_{2}^{1}&a_{3}^{1}\\\ 0&a_{1}^{2}&a_{2}^{2}&a_{3}^{2}\\\
0&a_{1}^{3}&a_{2}^{3}&a_{3}^{3}\end{matrix}\right)\left(\begin{matrix}\theta^{0}\\\
\theta^{1}\\\ \theta^{2}\\\
\theta^{3}\end{matrix}\right)=\left(\begin{matrix}1&0\\\
0&A\end{matrix}\right)\left(\begin{matrix}\theta^{0}\\\ \theta^{1}\\\
\theta^{2}\\\ \theta^{3}\end{matrix}\right),$

where the entries $a_{j}^{i}$ are functions on the jet space
$J^{2}\big{(}\mathbb{R}^{3},\mathbb{R}\big{)}$ and $\det A\neq 0$. Observe
that this transformation preserves the contact form $\theta^{0}$, as well as
the Pfaffian system $\big{\\{}\theta^{1},\theta^{2},\theta^{3}\big{\\}}$.
Denote by $\mathcal{A}$ the group consisting of such matrices222In [10],
Cartan allows for contact transformations on the full set of contact forms,
including shifts in the $\theta^{0}$ direction. Since the terms $\theta^{0}$
are not relevant to the symbol analysis as carried out in [25], we will
likewise fix these terms in the group of transformations we consider., $A$.
Then we will show presently that there is a prolongation
$\mathcal{A}^{\prime}$ of $\mathcal{A}$ that acts on the coframe
$\big{\\{}\omega^{i},\theta^{0},\theta^{i},\pi_{j}^{i}\big{\\}}$, $i=1,2,3,$
such that the form of the structure equations for the corresponding contact
system is preserved.

Let $A=(a_{j}^{i})$ and $A^{-1}=(\alpha_{j}^{i})$ so that
$\bar{\theta^{i}}=A\theta^{i}$ and $\theta^{i}=A^{-1}\bar{\theta^{i}}$. A
straightforward calculation using the definitions provided shows that

$\displaystyle{\rm
d}\bar{\theta}^{0}=\sum_{j=1}^{3}\bar{\omega}^{j}\wedge\bar{\theta}^{j},$

where $\bar{\omega}^{j}:=\sum\limits_{i=1}^{3}\alpha_{j}^{i}\omega^{i}$.
Likewise, utilizing the transformations
$\bar{\omega}=\big{(}A^{-1}\big{)}^{\rm T}\omega=\big{(}A^{\rm
T}\big{)}^{-1}\omega$ and $\omega=A^{\rm T}\bar{\omega}$, we can compute

$\displaystyle{\rm
d}\bar{\theta}^{i}\equiv\sum_{j=1}^{3}\bar{\omega}^{j}\wedge\bar{\pi}^{i}_{j}\quad\mod\big{\\{}\bar{\theta}^{i}\big{\\}},$

where
$\bar{\pi}_{j}^{i}:=\sum\limits_{k,l=1}^{3}a_{k}^{i}a_{l}^{j}\pi_{l}^{k}.$
Hence, if $\mathcal{A}^{\prime}$ acts on $\omega^{i}$ and $\pi_{j}^{i}$ as
follows

$\displaystyle\omega^{i}\longmapsto\sum_{j=1}^{3}\alpha_{i}^{j}\omega^{j}\qquad\textrm{and}\qquad\pi_{j}^{i}\longmapsto\sum_{k,l=1}^{3}a_{k}^{i}a_{l}^{j}\pi_{l}^{k},$

then the structure equations ${\rm
d}\theta^{i}=\sum\limits_{j=1}^{3}\omega^{j}\wedge\pi_{j}^{i}$ become the
congruences

$\displaystyle{\rm
d}\bar{\theta}^{i}\equiv\sum_{j=1}^{3}\bar{\omega}^{j}\wedge\bar{\pi}_{j}^{i}\quad\mod\big{\\{}\theta^{i}\big{\\}}$

and the following lemma may be stated.

######  Lemma A.1.

$\mathcal{A}^{\prime}$ prolongs $\mathcal{A}$, maps linear combinations of the
$\omega^{i}$ to linear combinations of $\omega^{i}$, and transforms the
structure equations (A.1) to

$\displaystyle{\rm
d}\bar{\theta}^{0}=\sum_{j=1}^{3}\bar{\omega}^{j}\wedge\bar{\theta}^{j},\qquad{\rm
d}\bar{\theta}^{i}\equiv\sum_{j=1}^{3}\bar{\omega}^{j}\wedge\bar{\pi}^{i}_{j}\quad\mod\bar{\theta}^{i},\qquad
i=1,2,3.$

Moreover, the $\pi_{j}^{i}$ are transformed as the coefficients of a quadratic
form.

The group $\mathcal{A}^{\prime}$ will later be utilized to simplify the
possible structure equations which must be considered in the structural
classification.

We will now consider more specifically systems of three second order
equations, $\Delta^{a}(x^{i},u,u_{i}$, $u_{ij})=0$, $1\leq a\leq 3$, which
satisfy the maximal rank condition that the $3\times 13$ Jacobian matrix

$\displaystyle
J_{\Delta}(x,u,u_{i},u_{ij})=\left(\frac{\partial\Delta^{a}}{\partial
x^{i}},\frac{\partial\Delta^{a}}{\partial u_{I}}\right)$

of $\Delta^{a}$ with respect to $x^{i}$, $u$, $u_{i}$, and $u_{ij}$ is of rank
3 whenever $\Delta^{a}=0$. To further ensure that the PDEs system is non-
degenerate, we require the $(3\times 6)$ submatrix

$\displaystyle\left(\frac{\partial\Delta^{a}}{\partial u_{ij}}\right)$

to have rank 3. Let $\mathcal{R}$ be the codimension 3 submanifold of
$J^{2}\big{(}\mathbb{R}^{3},\mathbb{R}\big{)}$ thus defined by $\Delta^{a}=0$,
and $\mathcal{C}^{2}(\mathcal{R})$ the corresponding contact system restricted
to $\mathcal{R}$. When the one forms
$\big{\\{}\omega^{i},\theta^{0},\theta^{i},\pi_{j}^{i}\big{\\}}$ are
restricted to $\mathcal{R}$, we obtain 3 linear relations between them. By
replacing $\pi_{j}^{i}$ with $\pi_{j}^{i}$ plus an appropriate linear
combination of the $\theta^{i}$, including $\theta^{0}$, these linear
relations may be simplified to the following

$\displaystyle\sum_{i,j=1}^{3}A_{i}^{jk}\pi_{j}^{i}+\sum_{l=1}^{3}A_{l}^{k}\omega^{l}=0\qquad\textrm{for}\quad
k=1,2,3.$ (A.2)

We will henceforth assume that the Pfaffian system
$\mathcal{C}^{2}(\mathcal{R})$ is an involution on which

$\displaystyle\omega^{1}\wedge\omega^{2}\wedge\omega^{3}\neq 0,$

in other words, there exists at least one Kähler-regular integral element of
the form

$\displaystyle\theta^{i}=0,\qquad\pi_{j}^{i}-\sum_{l=1}^{3}\alpha_{jl}^{i}\omega^{l}=0,$

where the $\alpha_{jl}^{i}$ are symmetric with respect to all indices. This
allows the relations (A.2) to be further reduced to

$\displaystyle\sum_{i,j=1}^{3}A_{i}^{jk}\pi_{j}^{i}=0\qquad\textrm{for}\quad
k=1,2,3.$ (A.3)

Finally, when considering the general 3-dimensional involution defined by

$\displaystyle\theta^{i}=0,\qquad\pi_{j}^{i}-\sum_{l=1}^{3}a_{jl}^{i}\omega^{l}=0,$

the relations (A.3) produce 9 equations

$\displaystyle\sum_{i,j=1}^{3}A_{i}^{jk}a_{jl}^{i}=0\qquad\textrm{for}\quad
l,k=1,2,3.$ (A.4)

We would like to know how many of the coefficients $a_{jl}^{i}$ are left
arbitrary by these 9 equations since Cartan’s involutivity test tells us that
$\mathcal{C}^{2}(\mathcal{R})$ is involutive with
$\omega^{1}\wedge\omega^{2}\wedge\omega^{3}\neq 0$ if and only if the number
of arbitrary coefficients $a_{jl}^{i}$ above equals $3\cdot
3-2\sigma_{1}-\sigma_{2}$ where $\sigma_{1}$ and $\sigma_{2}$ are the first
two reduced Cartan characters of $\mathcal{C}^{2}(\mathcal{R})$. Given that we
are concerned with the case of systems of 3 equations in 3 independent
variables, we can see that the Cartan characters $\sigma_{1}$ and $\sigma_{2}$
satisfy $\sigma_{1}\leq 3$ and $\sigma_{1}+\sigma_{2}\leq 3$. Thus the number
of arbitrary coefficients $a_{jl}^{i}$ will be $9-2\sigma_{1}-\sigma_{2}\geq
9-6=3$. Since the $a_{jl}^{i}$ are symmetric with respect to all indices,
there are 10 terms, which we now see are related by at most $10-3=7$
independent linear equations. Hence, at most 7 of the 9 equations (A.4) are
linearly independent. Making the identifications
$\pi_{j}^{i}\longleftrightarrow\hat{\xi}^{i}\hat{\xi}^{j}$ and
$a_{jl}^{i}\longleftrightarrow\hat{\xi}^{i}\hat{\xi}^{j}\hat{\xi}^{l}$, we can
state the following result:

######  Lemma A.2.

Defining the quadratic forms

$\displaystyle
M^{k}=\sum_{i,j=1}^{3}A_{i}^{jk}\hat{\xi}^{i}\hat{\xi}^{j},\qquad k=1,2,3,$

the $9$ cubic forms $\hat{\xi}^{i}M^{k}$ are related by at least $9-7=2$
linear equations.

The lemma states that the 9 cubic forms $\hat{\xi}^{i}M^{k}$ are related by at
least two linear relations, thus there are 6 linear forms, $l^{i}$ and
$m^{i}$, $i=1,2,3$, in the variables $\hat{\xi}^{i}$, such that these two
linear relations can be expressed by the two equations,

$\displaystyle l^{1}M^{1}+l^{2}M^{2}+l^{3}M^{3}=0,\qquad
m^{1}M^{1}+m^{2}M^{2}+m^{3}M^{3}=0.$

Given that these relations among the $M^{i}$ exist, one may take a linear
combination,

$\displaystyle\sum_{i=1}^{3}\big{(}z_{1}l^{i}+z_{2}m^{i}\big{)}M^{i}=0,$

and find $z_{1}$ and $z_{2}$ such that one of the coefficients
$z_{1}l^{i}+z_{2}m^{i}$ vanishes. In other words, we can choose $z_{1}$ and
$z_{2}$ such that the three linear forms $z_{1}l^{i}+z_{2}m^{i}$ are linearly
dependent. Equivalently, consider $z_{1}$ and $z_{2}$ as homogeneous
coordinates on the Riemann sphere, $\mathbb{P}_{1}(\mathbb{C})$, and define
$z:=z_{2}/z_{1}$. Then we have three linear forms $l^{i}+zm^{i}$, each of
which is a linear combination of the three variables $\hat{\xi}^{i}$. Taking
the coefficient of each of the three variables $\hat{\xi}^{i}$ in each of the
three expressions $l^{i}+zm^{i}$ yields a $3\times 3$ matrix. Setting the
determinant of this matrix equal to zero will determine $z$ in such a way that
the three linear forms $l^{i}+zm^{i}$ are linearly dependent, and produce a
cubic equation in $z$ with three roots in $\mathbb{P}_{1}(\mathbb{C})$.

To make this more explicit, and simplified, the group $\mathcal{A}^{\prime}$
may be used to induce linear transformations of the variables $\hat{\xi}^{i}$
which then induce corresponding linear transformations of $z_{1}$ and $z_{2}$.
This allows us to assume, for example, that one root of the equation mentioned
above is $z=0$. Then $l^{1}$, $l^{2},$ and $l^{3}$ are linearly dependent and
can be taken to be

$\displaystyle l^{1}=\hat{\xi}^{1},\qquad l^{2}=\hat{\xi}^{2},\qquad l^{3}=0.$

Furthermore, if we specify the $m^{i}$ by

$\displaystyle
m^{1}=a^{1}\hat{\xi}^{1}+b^{1}\hat{\xi}^{2}+c^{1}\hat{\xi}^{3},\qquad
m^{2}=a^{2}\hat{\xi}^{1}+b^{2}\hat{\xi}^{2}+c^{2}\hat{\xi}^{3},\qquad
m^{3}=a^{3}\hat{\xi}^{1}+b^{3}\hat{\xi}^{2}+c^{3}\hat{\xi}^{3},$

then the equation which $z$ must satisfy is the cubic polynomial given by,

$\displaystyle\det\left(\begin{matrix}1+a^{1}z&b^{1}z&c^{1}z\\\
a^{2}z&1+b^{2}z&c^{2}z\\\
a^{3}z&b^{3}z&c^{3}z\end{matrix}\right)=z\det\left(\begin{matrix}1+a^{1}z&b^{1}z&c^{1}z\\\
a^{2}z&1+b^{2}z&c^{2}z\\\ a^{3}&b^{3}&c^{3}\end{matrix}\right)=0.$ (A.5)

The structural classification then breaks into 5 possible cases, according to
whether the equation (A.5) has $(i)$ 3 simple roots, $(ii)$ one double and one
simple root, $(iii)$ one triple root, or $(iv)$ vanishes identically. Case
$(iv)$ splits further into two sub-cases according to which of the following
two forms the matrix in equation (A.5) can be reduced,

$\displaystyle\left(\begin{matrix}1&0&0\\\ 0&1&z\\\
1&0&0\end{matrix}\right)\qquad\textrm{or}\qquad\left(\begin{matrix}1&z&0\\\
0&1&0\\\ 1&0&0\end{matrix}\right).$

Hence there are 5 possible contact invariant structures in total, as described
in the following theorem.

######  Theorem A.3.

There are $5$ distinct classes of involutive systems of $3$ PDEs in $3$
independent and $1$ dependent variable. These $5$ classes are described
according to which of the $\pi_{j}^{i}$ vanish:

$\displaystyle(i)$ $\displaystyle\pi_{2}^{1}=\pi_{3}^{1}=\pi_{3}^{2}=0,$
$\displaystyle(ii)$ $\displaystyle\pi_{1}^{1}=\pi_{3}^{1}=\pi_{3}^{2}=0,$
$\displaystyle(iii)\ $
$\displaystyle\pi_{1}^{1}=\pi_{2}^{1}=\pi_{3}^{1}-\pi_{2}^{2}=0,$
$\displaystyle(iv)$ $\displaystyle\pi_{1}^{1}=\pi_{2}^{1}=\pi_{2}^{2}=0,$
$\displaystyle(v)$ $\displaystyle\pi_{1}^{1}=\pi_{2}^{1}=\pi_{3}^{1}=0.$

Finally, we will demonstrate to which of the five classes given in Theorem A.3
systems of the type (1.1) belong. Recall that the system of three equations
(1.1) is given by

$\displaystyle
F_{ij}\big{(}x^{1},x^{2},x^{3},u,u_{i},u_{j},u_{ij}\big{)}=u_{ij}-f_{ij}\big{(}x^{1},x^{2},x^{3},u,u_{i},u_{j}\big{)}=0\qquad\textrm{for}\quad
1\leq i<j\leq 3.$

Then by Lemma A.2 there exist at least 2 linear relations between the 3
quadratic forms

$\displaystyle M^{1}=\hat{\xi}^{1}\hat{\xi}^{2},\qquad
M^{2}=\hat{\xi}^{2}\hat{\xi}^{3},\qquad M^{3}=\hat{\xi}^{1}\hat{\xi}^{3}.$

Write these equations as

$\displaystyle
l^{1}\hat{\xi}^{1}\hat{\xi}^{2}+l^{2}\hat{\xi}^{2}\hat{\xi}^{3}+l^{3}\hat{\xi}^{1}\hat{\xi}^{3}=0,$
$\displaystyle
m^{1}\hat{\xi}^{1}\hat{\xi}^{2}+m^{2}\hat{\xi}^{2}\hat{\xi}^{3}+m^{3}\hat{\xi}^{1}\hat{\xi}^{3}=0.$

Then, for example, we can take
$\big{(}l^{1},l^{2},l^{3}\big{)}=\big{(}\hat{\xi}^{3},-\hat{\xi}^{1},0\big{)}$
and
$\big{(}m^{1},m^{2},m^{3}\big{)}=\big{(}\hat{\xi}^{3},\hat{\xi}^{1},-2\hat{\xi}^{2}\big{)}$
and set the following determinant equal to zero

$\displaystyle\det\left(\begin{matrix}0&0&1+z\\\ -1+z&0&0\\\
0&-2z&0\end{matrix}\right)=0.$

In other words, the equation $(1+z)(1-z)(2z)=0$ must hold. Here we have 3
simple roots, so the structure of this system fits into case $(i)$ of Theorem
A.3. The corresponding structure equations are then

$\displaystyle{\rm d}\theta^{0}\equiv 0,$ $\displaystyle{\rm
d}\theta^{1}\equiv\omega^{1}\wedge\pi_{1}^{1}\quad\mod\big{(}\theta^{0},\theta^{1},\theta^{2},\theta^{3}\big{)},$
$\displaystyle{\rm d}\theta^{2}\equiv\omega^{2}\wedge\pi_{2}^{2},$
$\displaystyle{\rm d}\theta^{3}\equiv\omega^{3}\wedge\pi_{3}^{3}.$

Cartan’s test then establishes that systems of the form (1.1) which satisfy
the integrability conditions $D_{k}u_{ij}=D_{i}u_{kj},\;\textrm{for}\;1\leq
i\neq j\neq k\leq 3$, are involutive and their solutions are parametrized by
three arbitrary functions of one variable.

##  Appendix B Structure equations and Lie bracket congruences

######  Proposition B.1.

The ${\rm d}_{H}$ structure equations for the Laplace adapted coframe (3.35)
for a system of the form (1.1) of three hyperbolic equations with commuting
characteristics are given by the following equations. Recall that
$p_{1}=\min\\{p_{21},p_{31}\\}$, $p_{2}=\min\\{p_{12},p_{32}\\}$, and
$p_{3}=\min\\{p_{13},p_{23}\\}$. Furthermore, $p_{k1}$, $p_{k2}$, and $p_{k3}$
will be used to denote which Laplace index achieves the minimum in the
definitions of $p_{1}$, $p_{2}$, and $p_{3}$ respectively; whereas $p_{l1}$,
$p_{l2}$, and $p_{l3}$ will denote the Laplace index which does not achieve
the minimum in the definitions of $p_{1}$, $p_{2}$, and $p_{3}$. So, for
example, if $p_{1}=p_{31}$ then $p_{k1}=p_{31}$ and $p_{l1}=p_{21}$,

$\displaystyle{\rm d}_{H}\sigma_{i}=0,$ (B.1) $\displaystyle{\rm
d}_{H}\Theta=\sigma_{1}\wedge\big{(}\hat{\xi}_{1}^{1}-A_{i1}^{i}\Theta\big{)}+\sigma_{2}\wedge\big{(}\hat{\xi}_{2}^{1}-A_{i2}^{i}\Theta\big{)}+\sigma_{3}\wedge\big{(}\hat{\xi}_{3}^{1}-A_{i3}^{i}\Theta\big{)},$
(B.2) $\displaystyle{\rm
d}_{H}\hat{\xi}_{1}^{1}=\sigma_{1}\wedge\big{(}\hat{\xi}_{1}^{2}-\big{(}A_{i1}^{i}\big{)}^{1}\hat{\xi}_{1}^{1}\big{)}+\sigma_{2}\wedge\big{(}H_{21}^{0}\Theta-
A_{21}^{1}\hat{\xi}_{1}^{1}\big{)}+\sigma_{3}\wedge\big{(}H_{31}^{0}\Theta-
A_{31}^{1}\hat{\xi}_{1}^{1}\big{)},$ (B.3)
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle{\rm
d}_{H}\hat{\xi}_{1}^{n}=\sigma_{1}\wedge\big{(}\hat{\xi}_{1}^{n+1}-\mathcal{X}_{1}^{n}\big{(}A_{i1}^{i}\big{)}\hat{\xi}_{1}^{n}\big{)}+\sigma_{2}\wedge\big{(}H_{21}^{n-1}\hat{\xi}_{1}^{n-1}-\mathcal{X}_{1}^{n-1}\big{(}A_{21}^{1}\big{)}\hat{\xi}_{1}^{n}\big{)}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{1}^{n}=}{}+\sigma_{3}\wedge\big{(}H_{31}^{n-1}\hat{\xi}_{1}^{n-1}-\mathcal{X}_{1}^{n-1}\big{(}A_{31}^{1}\big{)}\hat{\xi}_{1}^{n}\big{)}\qquad\textrm{for
all}\quad n,\quad 2\leq n\leq p_{k1},$ (B.4) $\displaystyle{\rm
d}_{H}\hat{\xi}_{1}^{n}=\sigma_{1}\wedge\big{(}\hat{\xi}_{1}^{n+1}-\big{(}A_{i1}^{i}\big{)}^{n}\hat{\xi}_{1}^{n}\big{)}+\sigma_{k}\wedge\big{(}{-}\big{(}A_{k1}^{1}\big{)}^{n-1}\hat{\xi}_{1}^{n}\big{)}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{1}^{n}=}{}+\sigma_{l}\wedge\big{(}H_{l1}^{n-1}\hat{\xi}_{1}^{n-1}-\big{(}A_{l1}^{1}\big{)}^{n-1}\hat{\xi}_{1}^{n}\big{)}\qquad\textrm{for}\quad
n=p_{k1}+1,$ (B.5) $\displaystyle{\rm
d}_{H}\hat{\xi}_{1}^{n}\equiv\sigma_{1}\wedge\big{(}\hat{\xi}_{1}^{n+1}-\big{(}A_{i1}^{i}\big{)}^{n}\hat{\xi}_{1}^{n}\big{)}+\sigma_{k}\wedge\big{(}{-}\big{(}A_{k1}^{1}\big{)}^{p_{k1}}\hat{\xi}_{1}^{n}\big{)}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{1}^{n}\equiv}{}+\sigma_{l}\wedge\big{(}H_{l1}^{n-1}\hat{\xi}_{1}^{n-1}-\big{(}A_{l1}^{1}\big{)}^{n-1}\hat{\xi}_{1}^{n}\big{)}\quad\mod\big{\\{}\hat{\xi}_{1}^{p_{k1}+1},\ldots,\hat{\xi}_{1}^{n-1}\big{\\}}$
$\displaystyle\hphantom{{\rm d}_{H}\hat{\xi}_{1}^{n}\equiv}{}\textrm{for}\quad
p_{k1}+2\leq n\leq p_{l1},$ (B.6) $\displaystyle{\rm
d}_{H}\hat{\xi}_{1}^{n}\equiv\sigma_{1}\wedge\hat{\xi}_{1}^{n+1}+\sigma_{k}\wedge\big{(}{-}\big{(}A_{k1}^{1}\big{)}^{p_{k1}}\hat{\xi}_{1}^{n}\big{)}+\sigma_{l}\wedge\big{(}{-}\big{(}A_{l1}^{1}\big{)}^{n-1}\hat{\xi}_{1}^{n}\big{)}$
$\displaystyle\hphantom{{\rm d}_{H}\hat{\xi}_{1}^{n}\equiv}{}\textrm{for}\quad
n=p_{l1}+1=p_{1}+1\quad\mod\big{\\{}\hat{\xi}_{1}^{p_{k1}+1},\ldots,\hat{\xi}_{1}^{n-1}\big{\\}},$
(B.7) $\displaystyle{\rm
d}_{H}\hat{\xi}_{1}^{n}\equiv\sigma_{1}\wedge\hat{\xi}_{1}^{n+1}+\sigma_{k}\wedge\big{(}{-}\big{(}A_{k1}^{1}\big{)}^{p_{k1}}\hat{\xi}_{1}^{n})+\sigma_{l}\wedge\big{(}{-}\big{(}A_{l1}^{1}\big{)}^{p_{l1}}\hat{\xi}_{1}^{n}\big{)}$
$\displaystyle\hphantom{{\rm d}_{H}\hat{\xi}_{1}^{n}\equiv}{}\textrm{for}\quad
n\geq
p_{1}+2\quad\mod\big{\\{}\hat{\xi}_{1}^{p_{k1}+2},\ldots,\hat{\xi}_{1}^{n-1}\big{\\}},$
(B.8)

where $p_{ij}$ is the smallest integer such that $H_{ij}^{p_{ij}}=0$. Similar
formulas hold for the forms $\hat{\xi}_{2}^{i}$ and $\hat{\xi}_{3}^{i}$:

$\displaystyle{\rm
d}_{H}\hat{\xi}_{2}^{1}=\sigma_{1}\wedge\big{(}H_{12}^{0}\Theta-
A_{12}^{2}\hat{\xi}_{2}^{1}\big{)}+\sigma_{2}\wedge\big{(}\hat{\xi}_{2}^{2}-\big{(}A_{i2}^{i}\big{)}^{1}\hat{\xi}_{2}^{1}\big{)}+\sigma_{3}\wedge\big{(}H_{32}^{0}\Theta-
A_{32}^{2}\hat{\xi}_{2}^{1}\big{)},$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle{\rm
d}_{H}\hat{\xi}_{2}^{n}=\sigma_{1}\wedge\big{(}H_{12}^{n-1}\hat{\xi}_{2}^{n-1}-\big{(}A_{12}^{2}\big{)}^{n-1}\hat{\xi}_{2}^{n}\big{)}+\sigma_{2}\wedge\big{(}\hat{\xi}_{2}^{n+1}-\big{(}A_{i2}^{i}\big{)}^{n}\hat{\xi}_{2}^{n}\big{)}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{2}^{n}=}{}+\sigma_{3}\wedge\big{(}H_{32}^{n-1}\hat{\xi}_{2}^{n-1}-\big{(}A_{32}^{2}\big{)}^{n-1}\hat{\xi}_{2}^{n}\big{)}\qquad\textrm{for}\quad
2\leq n\leq p_{k2},$ $\displaystyle{\rm
d}_{H}\hat{\xi}_{2}^{n}=\sigma_{k}\wedge\big{(}{-}\big{(}A_{k2}^{2}\big{)}^{n-1}\hat{\xi}_{2}^{n}\big{)}+\sigma_{2}\wedge\big{(}\hat{\xi}_{2}^{n+1}-\big{(}A_{i2}^{i}\big{)}^{n}\hat{\xi}_{2}^{n}\big{)}+\sigma_{l}\wedge\big{(}H_{l2}^{n-1}\hat{\xi}_{2}^{n-1}-\big{(}A_{l2}^{2}\big{)}^{n-1}\hat{\xi}_{2}^{n}\big{)}$
$\displaystyle\hphantom{{\rm d}_{H}\hat{\xi}_{2}^{n}=}{}\textrm{for}\quad
n=p_{k2}+1,$ $\displaystyle{\rm
d}_{H}\hat{\xi}_{2}^{n}\equiv\sigma_{k}\wedge\big{(}{-}\big{(}A_{k2}^{2}\big{)}^{p_{k2}}\hat{\xi}_{2}^{n}\big{)}+\sigma_{2}\wedge\big{(}\hat{\xi}_{2}^{n+1}-\big{(}A_{i2}^{i}\big{)}^{n}\hat{\xi}_{2}^{n}\big{)}+\sigma_{l}\wedge\big{(}H_{l2}^{n-1}\hat{\xi}_{2}^{n-1}-\big{(}A_{l2}^{2}\big{)}^{n-1}\hat{\xi}_{2}^{n}\big{)}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{2}^{n}\equiv}{}\mod\big{\\{}\hat{\xi}_{2}^{p_{k2}+1},\ldots,\hat{\xi}_{2}^{n-1}\big{\\}}\qquad\textrm{for}\quad
p_{k2}+2\leq n\leq p_{l2},$ $\displaystyle{\rm
d}_{H}\hat{\xi}_{2}^{n}\equiv\sigma_{k}\wedge\big{(}{-}\big{(}A_{k2}^{2}\big{)}^{p_{k2}}\hat{\xi}_{2}^{n}\big{)}+\sigma_{2}\wedge\hat{\xi}_{2}^{n+1}+\sigma_{l}\wedge\big{(}{-}\big{(}A_{l2}^{2}\big{)}^{n-1}\hat{\xi}_{2}^{n}\big{)}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{2}^{n}\equiv}{}\mod\big{\\{}\hat{\xi}_{2}^{p_{k2}+1},\ldots,\hat{\xi}_{2}^{n-1}\big{\\}}\qquad\textrm{for}\quad
n\geq p_{l2}+1=p_{2}+1$

and

$\displaystyle{\rm
d}_{H}\hat{\xi}_{3}^{1}=\sigma_{1}\wedge\big{(}H_{13}^{0}\Theta-
A_{13}^{3}\hat{\xi}_{3}^{1}\big{)}+\sigma_{2}\wedge\big{(}H_{23}^{0}\Theta-
A_{23}^{3}\hat{\xi}_{3}^{1}\big{)}+\sigma_{3}\wedge\big{(}\hat{\xi}_{3}^{2}-\big{(}A_{i3}^{i}\big{)}^{1}\hat{\xi}_{3}^{1}\big{)},$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle{\rm
d}_{H}\hat{\xi}_{3}^{n}=\sigma_{1}\wedge\big{(}H_{13}^{n-1}\hat{\xi}_{3}^{n-1}-\big{(}A_{13}^{3}\big{)}^{n-1}\hat{\xi}_{3}^{n}\big{)}+\sigma_{2}\wedge\big{(}H_{23}^{n-1}\hat{\xi}_{3}^{n-1}-\big{(}A_{23}^{3}\big{)}^{n-1}\hat{\xi}_{3}^{n}\big{)}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{3}^{n}=}{}+\sigma_{3}\wedge\big{(}\hat{\xi}_{3}^{n+1}-\big{(}A_{i3}^{i}\big{)}^{n}\hat{\xi}_{3}^{n}\big{)}\qquad\textrm{for}\quad
2\leq n\leq p_{k3},$ $\displaystyle{\rm
d}_{H}\hat{\xi}_{3}^{n}=\sigma_{k}\wedge\big{(}{-}\big{(}A_{k3}^{3}\big{)}^{n-1}\hat{\xi}_{3}^{n}\big{)}+\sigma_{l}\wedge\big{(}H_{l3}^{n-1}-\big{(}A_{l3}^{3}\big{)}^{n-1}\hat{\xi}_{3}^{n}\big{)}+\sigma_{3}\wedge\big{(}\hat{\xi}_{3}^{n+1}-\big{(}A_{i3}^{i}\big{)}^{n}\hat{\xi}_{3}^{n}\big{)}$
$\displaystyle\hphantom{{\rm d}_{H}\hat{\xi}_{3}^{n}=}{}\textrm{for}\quad
n=p_{k3}+1,$ $\displaystyle{\rm
d}_{H}\hat{\xi}_{3}^{n}\equiv\sigma_{k}\wedge\big{(}{-}\big{(}A_{k3}^{3}\big{)}^{p_{k3}}\hat{\xi}_{3}^{n}\big{)}+\sigma_{l}\wedge\big{(}H_{l3}^{n-1}\hat{\xi}_{3}^{n-1}-\big{(}A_{l3}^{3}\big{)}^{n-1}\hat{\xi}_{3}^{n}\big{)}+\sigma_{3}\wedge\big{(}\hat{\xi}_{3}^{n+1}-\big{(}A_{i3}^{i}\big{)}^{n}\hat{\xi}_{3}^{n}\big{)}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{3}^{n}\equiv}{}\mod\big{\\{}\hat{\xi}_{3}^{p_{k3}+1},\ldots,\hat{\xi}_{3}^{n-1}\big{\\}}\qquad\textrm{for}\quad
p_{k3}+2\leq n\leq p_{l3},$ $\displaystyle{\rm
d}_{H}\hat{\xi}_{3}^{n}\equiv\sigma_{k}\wedge\big{(}{-}\big{(}A_{k3}^{3}\big{)}^{p_{k3}}\hat{\xi}_{3}^{n}\big{)}+\sigma_{l}\wedge\big{(}{-}\big{(}A_{l3}^{3}\big{)}^{n-1}\hat{\xi}_{3}^{n}\big{)}+\sigma_{3}\wedge\hat{\xi}_{3}^{n+1}$
$\displaystyle\hphantom{{\rm
d}_{H}\hat{\xi}_{3}^{n}\equiv}{}\mod\big{\\{}\hat{\xi}_{3}^{p_{k3}+1},\ldots,\hat{\xi}_{3}^{n-1}\big{\\}}\qquad\textrm{for}\quad
n\geq p_{l3}+1=p_{3}+1.$

###### Proof.

The structure equations ${\rm d}_{H}\sigma_{i}=0$ follow from the assumption
that the characteristic vector fields $X_{i}$ all commute, along with
Definition 2.3 which implies that $({\rm d}_{V}\sigma_{i})(X_{i},X_{j})=0$.
Thus

$\displaystyle({\rm d}_{H}\sigma_{i})(X_{i},X_{j})=({\rm
d}\sigma_{i})(X_{i},X_{j})=-\sigma_{i}([X_{i},X_{j}])=0$

for any pair of $X_{i}$ and $X_{j}$. The other structure equations are
computed by using the formula

$\displaystyle{\rm d}_{H}\omega=\sigma_{1}\wedge
X_{1}(\omega)+\sigma_{2}\wedge X_{2}(\omega)+\sigma_{3}\wedge X_{3}(\omega).$
(B.9)

The formula for ${\rm d}_{H}\Theta$ follows immediately from the definitions
of $\hat{\xi}_{1}^{1}$, $\hat{\xi}_{1}^{2}$, and $\hat{\xi}_{1}^{3}$. The
remaining formulas are obtained by using equations (3.36)–(3.39) in formula
(B.9). For example, for ${\rm d}_{H}\hat{\xi}_{1}^{1}$ we use equation (3.36)
with $n=2$ and $j=1$ to see that

$\displaystyle\sigma_{1}\wedge
X_{1}\big{(}\hat{\xi}_{1}^{1}\big{)}=\sigma_{1}\wedge\big{(}\hat{\xi}_{1}^{2}-\big{(}A_{i1}^{i}\big{)}^{1}\hat{\xi}_{1}^{1}\big{)}.$

Equation (3.37) with $i=2$ and $j=1$ shows that

$\displaystyle\sigma_{2}\wedge
X_{2}\big{(}\hat{\xi}_{1}^{1}\big{)}=\sigma_{2}\wedge\big{(}H_{21}^{0}\Theta-\big{(}A_{21}^{1}\big{)}\hat{\xi}_{1}^{1}\big{)}$

and likewise, (3.37) with $i=3$ and $j=1$ gives $\sigma_{3}\wedge
X_{3}\big{(}\hat{\xi}_{1}^{1}\big{)}=\sigma_{3}\wedge\big{(}H_{31}^{0}\Theta-\big{(}A_{31}^{1}\big{)}\hat{\xi}_{1}^{1}\big{)}$.
Similar substitutions yields the ${\rm d}_{H}$ structure formulas (B.4)–(B.7)
above. Finally, formula (B.8) is proved by induction on $n$, beginning with

$\displaystyle
X_{k}\big{(}\hat{\xi}_{1}^{n+1}\big{)}=X_{k}\big{(}X_{1}\big{(}\hat{\xi}_{1}^{n}\big{)}\big{)}=X_{1}\big{(}X_{k}\big{(}\hat{\xi}_{1}^{n}\big{)}\big{)}$
$\displaystyle\hphantom{X_{k}\big{(}\hat{\xi}_{1}^{n+1}\big{)}}{}\equiv
X_{1}\big{(}{-}\big{(}A_{k1}^{1}\big{)}^{p_{k1}}\hat{\xi}_{1}^{n}\big{)}\quad\mod\big{\\{}\hat{\xi}_{1}^{p_{k1}+2},\ldots,\hat{\xi}_{1}^{n-1}\big{\\}},\qquad\textrm{for}\quad
k=2,3,$

where the above equalities are obtained by making use of equation (3.39). The
sets of ${\rm d}_{H}$ structure formulas for $\hat{\xi}_{2}^{n}$ and
$\hat{\xi}_{3}^{n}$ are derived via an analogous procedure, again using the
appropriate versions of equations (3.36)–(3.39) in formula (B.9). ∎

The Lie bracket congruences for the characteristic vector fields $X_{i}$ with
the vertical vector fields $U$ and $V_{k}^{l}$ defined by

$\displaystyle\Theta(U)=1,\qquad\Theta\big{(}V_{k}^{l}\big{)}=0,\qquad\hat{\xi}_{j}^{n}(U)=0,\qquad\hat{\xi}_{j}^{n}\big{(}V_{k}^{l}\big{)}=\delta_{k}^{j}\delta_{l}^{n}$

are given in the following proposition.

######  Proposition B.2.

The following Lie bracket congruences hold modulo $\\{X_{1},X_{2},X_{3}\\}$:

$\displaystyle[X_{1},U]\equiv
A_{i1}^{i}U-H_{12}^{0}V_{2}^{1}-H_{13}^{0}V_{3}^{1}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
(B.10)
$\displaystyle\big{[}X_{1},V_{1}^{1}\big{]}\equiv-U+\big{(}A_{i1}^{i}\big{)}^{1}V_{1}^{1}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
(B.11) $\displaystyle\big{[}X_{1},V_{1}^{n}\big{]}\equiv-
V_{1}^{n-1}+\big{(}A_{i1}^{i}\big{)}^{n}V_{1}^{n}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n\leq p_{l1},$ (B.12) $\displaystyle\big{[}X_{1},V_{1}^{n}\big{]}\equiv-
V_{1}^{n-1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad n\geq
p_{l1}+1,$ $\displaystyle\big{[}X_{1},V_{2}^{1}\big{]}\equiv
A_{12}^{2}V_{2}^{1}-H^{1}_{12}V_{2}^{2}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{1},V_{2}^{n}\big{]}\equiv\big{(}A_{12}^{2}\big{)}^{n-1}V_{2}^{n}-H_{12}^{n}V_{2}^{n+1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n<p_{12},$
$\displaystyle\big{[}X_{1},V_{2}^{n}\big{]}\equiv\big{(}A_{12}^{2}\big{)}^{p_{12}}V_{2}^{n}\quad\mod\big{\\{}X_{1},X_{2},X_{3},V_{2}^{n+1},\ldots\big{\\}}\qquad\textrm{for}\quad
n\geq p_{12},$ (B.13) $\displaystyle\big{[}X_{1},V_{3}^{1}\big{]}\equiv
A_{13}^{3}V_{3}^{1}-H^{1}_{13}V_{3}^{2}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
(B.14)
$\displaystyle\big{[}X_{1},V_{3}^{n}\big{]}\equiv\big{(}A_{13}^{3}\big{)}^{n-1}V_{3}^{n}-H_{13}^{n}V_{3}^{n+1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n<p_{13},$ (B.15)
$\displaystyle\big{[}X_{1},V_{3}^{n}\big{]}\equiv\big{(}A_{13}^{3}\big{)}^{p_{13}}V_{3}^{n}\quad\mod\big{\\{}X_{1},X_{2},X_{3},V_{3}^{n+1},\ldots\big{\\}}\qquad\textrm{for}\quad
n\geq p_{13},$ $\displaystyle[X_{2},U]\equiv
A_{i2}^{i}U-H_{21}^{0}V_{1}^{1}-H_{23}^{0}V_{3}^{1}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{2},V_{1}^{1}\big{]}\equiv
A_{21}^{1}V_{1}^{1}-H_{21}^{1}V_{1}^{2}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{2},V_{1}^{n}\big{]}\equiv\big{(}A_{21}^{1}\big{)}^{n-1}V_{1}^{n}-H_{21}^{n}V_{1}^{n+1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n<p_{21},$
$\displaystyle\big{[}X_{2},V_{1}^{n}\big{]}\equiv\big{(}A_{21}^{1}\big{)}^{p_{21}}V_{1}^{n}\quad\mod\big{\\{}X_{1},X_{2},X_{3},V_{1}^{n+1},\ldots\big{\\}}\qquad\textrm{for}\quad
n\geq p_{21},$
$\displaystyle\big{[}X_{2},V_{2}^{1}\big{]}\equiv-U+\big{(}A_{i2}^{i}\big{)}^{1}V_{2}^{1}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{2},V_{2}^{n}\big{]}\equiv-
V_{2}^{n-1}+\big{(}A_{i2}^{i}\big{)}^{n}V_{2}^{n}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n\leq p_{l2},$ $\displaystyle\big{[}X_{2},V_{2}^{n}\big{]}\equiv-
V_{2}^{n-1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad n\geq
p_{l2}+1,$ $\displaystyle\big{[}X_{2},V_{3}^{1}\big{]}\equiv
A_{23}^{3}V_{3}^{1}-H_{23}^{1}V_{3}^{2}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{2},V_{3}^{n}\big{]}\equiv\big{(}A_{23}^{3}\big{)}^{n-1}V_{3}^{n}-H_{23}^{n}V_{3}^{n+1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n<p_{23},$
$\displaystyle\big{[}X_{2},V_{3}^{n}\big{]}\equiv\big{(}A_{23}^{3}\big{)}^{p_{23}}V_{3}^{n}\quad\mod\big{\\{}X_{1},X_{2},X_{3},V_{3}^{n+1},\ldots\big{\\}}\qquad\textrm{for}\quad
n\geq p_{23},$ $\displaystyle[X_{3},U]\equiv
A_{i3}^{i}U-H_{31}^{0}V_{1}^{1}-H_{32}^{0}V_{2}^{1}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{3},V_{1}^{1}\big{]}\equiv
A_{31}^{1}V_{1}^{1}-H_{31}^{1}V_{1}^{2}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{3},V_{1}^{n}\big{]}\equiv\big{(}A_{31}^{1}\big{)}^{n-1}V_{1}^{n}-H_{31}^{n}V_{1}^{n+1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n<p_{31},$
$\displaystyle\big{[}X_{3},V_{1}^{n}\big{]}\equiv\big{(}A_{31}^{1}\big{)}^{p_{31}}V_{1}^{n}\quad\mod\big{\\{}X_{1},X_{2},X_{3},V_{1}^{n+1},\ldots\big{\\}}\qquad\textrm{for}\quad
n\geq p_{31},$ $\displaystyle\big{[}X_{3},V_{2}^{1}\big{]}\equiv
A_{32}^{2}V_{2}^{1}-H^{1}_{32}V_{2}^{2}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{3},V_{2}^{n}\big{]}\equiv\big{(}A_{32}^{2}\big{)}^{n-1}V_{2}^{n}-H_{32}^{n}V_{2}^{n+1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n<p_{32},$
$\displaystyle\big{[}X_{3},V_{2}^{n}\big{]}\equiv\big{(}A_{32}^{2}\big{)}^{p_{32}}V_{2}^{n}\quad\mod\big{\\{}X_{1},X_{2},X_{3},V_{2}^{n+1},\ldots\big{\\}}\qquad\textrm{for}\quad
n\geq p_{32},$
$\displaystyle\big{[}X_{3},V_{3}^{1}\big{]}\equiv-U+\big{(}A_{i3}^{i}\big{)}^{1}V_{3}^{1}\quad\mod\\{X_{1},X_{2},X_{3}\\},$
$\displaystyle\big{[}X_{3},V_{3}^{n}\big{]}\equiv-
V_{3}^{n-1}+\big{(}A_{i3}^{i}\big{)}^{n}V_{3}^{n}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad
2\leq n\leq p_{l3},$ $\displaystyle\big{[}X_{3},V_{3}^{n}\big{]}\equiv-
V_{3}^{n-1}\quad\mod\\{X_{1},X_{2},X_{3}\\}\qquad\textrm{for}\quad n\geq
p_{l3}+1.$

###### Proof.

If $X$ is a total vector field and $V$ is a $\pi_{M}^{\infty}$ vertical vector
field on $\mathcal{R}^{\infty}$ and $\omega$ is a contact 1-form, then the
following equation holds

$\displaystyle({\rm d}_{H}\omega)(X,V)=({\rm
d}\omega)(X,V)=X(\omega(V))-V(\omega(X))-w([X,V])=X(\omega(V))-\omega([X,V]),$

since $\omega(X)=0$ by definition of $X$ being a total vector field, and ${\rm
d}_{V}\omega(X,V)=0$ again by definition because ${\rm d}_{V}\omega$ is a type
$(0,2)$ contact form. Moreover, if $\omega(V)$ is a constant, then

$\displaystyle({\rm d}_{H}\omega)(X,V)=-\omega([X,V]).$

This last equation, along with the ${\rm d}_{H}$ structure equations from
Theorem B.1, allow us to obtain the above Lie bracket congruences. As an
example, let’s compute $[X_{1},U]$. Taking the structure equation (B.2), we
see that $\Theta([X_{1},U])=-({\rm d}_{H}\Theta)(X_{1},U)=A_{i1}^{i}$. This
tells us that the bracket $[X_{1},U]$ has a $U$ component with coefficient
$A_{i1}^{i}$. Likewise, taking equation (B.3) shows that
$\hat{\xi}_{1}^{1}([X_{1},U])=-\big{(}{\rm
d}_{H}\hat{\xi}_{1}^{1}\big{)}(X_{1},U)=0$, which implies that $[X_{1},U]$ has
no $V_{1}^{1}$ component. Continue on in this way using each of the remaining
${\rm d}_{H}\hat{\xi}_{i}^{j}$ structure equations in turn in order to
identify all the nonzero components of $[X_{1},U]$. Repeat the same procedure
with $[X_{2},U]$, $[X_{3},U]$, and $\big{[}X_{i},V_{k}^{l}\big{]}$ to obtain
all of the above Lie bracket congruences. ∎

### Acknowledgements

I would like to express my heartfelt gratitude to my Ph.D. advisor, Professor
Niky Kamran for his constant encouragement and guidance throughout the
preparation of this paper. I would also like to thank Professor Mark Fels for
the lectures he presented during the spring of 2012 at McGill University,
which contributed greatly to my understanding of Cartan’s structural
classification of involutive overdetermined systems of PDEs. And for bringing
to my attention the example presented in Section 3.1, I thank Professor Peter
Vassiliou. Finally, I thank the referees of this paper for their many
thoughtful and helpful comments.

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