# On uniqueness of submaximally symmetric
vector ordinary differential equations of C-class

Johnson Allen Kessy  Department of Mathematics and Statistics, UiT The Arctic
University of Norway, 9037 Tromsø, Norway<EMAIL_ADDRESS>and  Dennis
The  Department of Mathematics and Statistics, UiT The Arctic University of
Norway, 9037 Tromsø, Norway<EMAIL_ADDRESS>
###### Abstract.

The fundamental invariants for vector ODEs of order $\geq 3$ considered up to
point transformations consist of generalized Wilczynski invariants and C-class
invariants. An ODE of C-class is characterized by the vanishing of the former.
For any fixed C-class invariant $\mathcal{U}$, we give a local (point)
classification for all submaximally symmetric ODEs of C-class with
$\mathcal{U}\not\equiv 0$ and all remaining C-class invariants vanishing
identically. Our results yield generalizations of a well-known classical
result for scalar ODEs due to Sophus Lie.

Fundamental invariants correspond to the harmonic curvature of the associated
Cartan geometry. A key new ingredient underlying our classification results is
an advance concerning the harmonic theory associated with the structure of
vector ODEs of C-class. Namely, for each irreducible C-class module, we
provide an explicit identification of a lowest weight vector as a harmonic
2-cochain.

###### Key words and phrases:

Submaximal symmetry, System of ODEs, C-class equations, Cartan geometry

###### 2020 Mathematics Subject Classification:

Primary: 35B06, 53A55; Secondary: 17B66, 57M60.

##  1\. Introduction

Finite dimensionality of the contact symmetry algebra for scalar ODEs
$u_{n+1}=f(t,u,u_{1},\ldots,u_{n})$ of order $n+1\geq 4$ is a classical result
due to Sophus Lie [16] (see also [20, Thm 6.44]). (We use jet notation $u_{k}$
instead of the more standard notation $u^{(k)}$ to denote the $k$-th
derivative of $u$ with respect to $t$.) The maximal symmetry dimension and the
submaximal (i.e. next largest realizable) symmetry dimension are respectively:

(1.1)
$\displaystyle\mathfrak{M}:=n+5\qquad\mbox{and}\qquad\mathfrak{S}:=\begin{cases}\mathfrak{M}-1,&\mbox{for}\quad
n=4\quad\mbox{or}\quad 6;\\\ \mathfrak{M}-2,&\mbox{otherwise}.\end{cases}$

The former is realized locally uniquely by the trivial ODE $u_{n+1}=0$. For
ODEs realizing $\mathfrak{S}$, we have the following result (over
$\mathbb{C}$) due to Lie [17] (see also [20, pp. 205-206]): Any submaximally
symmetric scalar ODE of order $n+1\geq 4$ is locally contact-equivalent to:

  1. (a)

a linear equation, or

  2. (b)

exactly one of:111In [20, p. 206], the ODE $3u_{2}u_{4}-5(u_{3})^{2}=0$ is
also listed as a possibility, but this is in fact contact-equivalent to
$nu_{n-1}u_{n+1}-(n+1)(u_{n})^{2}=0$ when $n=3$. We have verified this using
Cartan-geometric techniques – details will be given elsewhere.

    1. (i)

$n=4$: $9(u_{2})^{2}u_{5}-45u_{2}u_{3}u_{4}+40(u_{3})^{3}=0$.

    2. (ii)

$n=6$:
$10(u_{3})^{3}u_{7}-70(u_{3})^{2}u_{4}u_{6}-49(u_{3})^{2}(u_{5})^{2}+280u_{3}(u_{4})^{2}u_{5}-175(u_{4})^{4}=0$.

    3. (iii)

$n\neq 4,6$: $nu_{n-1}u_{n+1}-(n+1)(u_{n})^{2}=0$.

The aim of our article is to establish analogous results for vector ODEs
$\mathcal{E}$ of order $n+1\geq 3$:

(1.2)
$\displaystyle\mathbf{u}_{n+1}=\mathbf{f}(t,\mathbf{u},\mathbf{u}_{1},\ldots,\mathbf{u}_{n}),$

where $\mathbf{u}$ is an $\mathbb{R}^{m}$-valued function of $t$ (for $m\geq
2$), and $\mathbf{u}_{k}$ is its $k$-th derivative. More precisely, we
consider and completely resolve the classification problem (up to local
contact equivalence) for submaximally symmetric vector ODEs (1.2) of order
$\geq 3$ of C-class [6, 4] (see below for motivation). Note that by the
Lie–Bäcklund theorem, contact-equivalence agrees with point-equivalence for
vector ODEs.

For vector ODEs (1.2) of order $n+1\geq 3$, the maximal and submaximal
symmetry dimensions are:

(1.3)
$\displaystyle\mathfrak{M}=m^{2}+(n+1)m+3\qquad\mbox{and}\qquad\mathfrak{S}=\mathfrak{M}-2,$

with the latter established in our earlier work [13], along with numerous
other symmetry gap results. The trivial vector ODE
$\mathbf{u}_{n+1}=\mathbf{0}$ is locally uniquely maximally symmetric – see
for example [13, Cor 2.8]. Examples of some submaximally symmetric vector ODEs
were given in [13, Table 8], but no definitive classification lists for the
submaximal strata were asserted. This is a focus of our current article.

Following Cartan [6] (see also [2, 12, 4]), a class of vector ODE (1.2) of
order $\geq 3$ is said to be a C-class if it is invariant under all contact
transformations, and all (contact) differential invariants of any ODE in this
class are first integrals of that ODE. Hence, generic C-class equations
(having sufficiently many functionally independent first integrals) can be
solved using these invariants. In [4, Thms 4.1 & 4.2], the C-class was
characterized by the vanishing of the generalized Wilczynski invariants. These
are a subset of the fundamental (relative) invariants, which additionally
consist of C-class invariants (in the terminology of [13]).

We note from [13, Tables 8 & 10] that a vector ODE realizing $\mathfrak{S}$
given in (1.3) is either a 3rd order ODE pair of C-class (i.e. $(n,m)=(2,2)$)
or it is of Wilczynski type (i.e. an ODE with all C-class invariants vanishing
identically). We will prove the following generalization of Lie’s result above
for vector ODEs:

######  Theorem 1.1.

Any submaximally symmetric vector ODE (1.2) of order $n+1\geq 3$ is either:

  1. (a)

of Wilczynski type, or

  2. (b)

locally equivalent222More precisely, “local equivalence” here is meant in a
neighbourhood of a point in $\mathcal{E}$ where at least one of the C-class
invariants is non-zero. over $\mathbb{R}$ to exactly one of the three 3rd
order ODE pairs in Table 1. Over $\mathbb{C}$, the two 3rd order ODE pairs in
the second row of Table 1 are locally equivalent.

Lie obtained his result for submaximally symmetric scalar ODEs using his
complete classification of Lie algebras of contact vector fields on the
(complex) plane and classified invariant ODEs having sufficiently many
symmetries. Certainly, this approach generalizes to vector ODEs, but it is not
feasible: complete classifications for Lie algebras of (point) vector fields
on $\mathbb{C}^{n}$ or $\mathbb{R}^{n}$ for $n\geq 3$ are known to be very
difficult to establish [9, 21]. So, different techniques are needed to
establish analogous results for submaximally symmetric vector ODEs.

Our approach to classifying all submaximally symmetric vector ODEs (1.2) of
C-class of order $\geq 3$ is motivated by that of [23, 24] in the setting of
parabolic geometries [5], and is based on a categorically equivalent
reformulation of vector ODEs (1.2) as (strongly) regular, normal Cartan
geometries $(\mathcal{G}\to\mathcal{E},\omega)$ of type $(G,P)$ for a certain
Lie group $G$ and closed subgroup $P\subset G$ [4, 10] (see §2.3 below).

For such a (non-parabolic) Cartan geometry, the harmonic curvature
$\kappa_{H}$, which corresponds to the fundamental invariants, is valued in a
certain $P$-module that is completely reducible [4, Cor 3.8], so only the
action of the reductive part $G_{0}\subset P$ is relevant. Via a known
algebraic Hodge theory associated with $G_{0}$, the codomain of $\kappa_{H}$
can be identified with a certain $G_{0}$-submodule $\mathbb{E}\subsetneq
H^{2}(\mathfrak{g}_{-},\mathfrak{g})$ of a Lie algebra cohomology group called
the effective part (see Definition 2.3). This has been already computed for
ODEs (1.2) of order $3$ in [18, 19] and of order $\geq 4$ in [11]. The
aforementioned fundamental invariants are valued in corresponding
$G_{0}$-irreducible submodules $\mathbb{U}\subset\mathbb{E}$; see [13, Table
6] for a summary. The irreducible C-class modules are listed in Table 2.

We next formulate our second main result, which concerns the classification of
vector ODEs (1.2) of C-class realizing the so-called constrained submaximal
symmetry dimensions $\mathfrak{S}_{\mathbb{U}}$ identified in [13, Table 2].
Fix an irreducible C-class module
$\mathbb{U}=\mathbb{B}_{4},\mathbb{A}_{2}^{\operatorname{tr}},\mathbb{A}_{2}^{\operatorname{tf}}\subset\mathbb{E}$
(see §2.4) and its corresponding C-class invariant
$\mathcal{U}=\mathcal{B}_{4},\mathcal{A}_{2}^{\operatorname{tr}},\mathcal{A}_{2}^{\operatorname{tf}}$
(see §2.5). Let $C_{\mathcal{U}}$ denote the set of all ODEs (1.2) with
$\mathcal{U}\not\equiv 0$ and all remaining C-class invariants vanishing
identically (equivalently,
$0\not\equiv\operatorname{im}(\kappa_{H})\subset\mathbb{U}$), and let
$\mathfrak{S}_{\mathbb{U}}$ denote the largest realizable symmetry dimension
among ODEs in $C_{\mathcal{U}}$. We will prove the following classification
result:

######  Theorem 1.2.

Any vector ODE (1.2) $\mathcal{E}$ of C-class of order $n+1\geq 3$ in
$C_{\mathcal{U}}$ realizing $\mathfrak{S}_{\mathbb{U}}$, near any point
$x\in\mathcal{E}$ with $\mathcal{U}(x)\neq 0$, is locally (point) equivalent
over $\mathbb{R}$ to exactly one of the ODEs given in Table 1. Over
$\mathbb{C}$, the indicated 3rd order ODEs for $\mathbb{U}=\mathbb{B}_{4}$ are
locally equivalent.

$\begin{array}[]{|cccc|}\hline\cr n&\begin{array}[]{c}\mbox{Irreducible
C-class}\\\
\mbox{module}\,\,\mathbb{U}\subset\mathbb{E}\end{array}&\mathfrak{S}_{\mathbb{U}}&\begin{array}[]{c}\mbox{
ODE of C-class
with}\,\,0\not\equiv\operatorname{im}(\kappa_{H})\subset\mathbb{U}\\\
\mbox{with symmetry dimension
realizing}\,\,\mathfrak{S}_{\mathbb{U}}\end{array}\\\ \hline\cr\hline\cr
2&\mathbb{B}_{4}&\mathfrak{M}-m&\begin{array}[]{c}\underset{(1\leq a\leq
m)}{u_{3}^{a}=\displaystyle\frac{3u_{2}^{1}u_{2}^{a}}{2u^{1}_{1}}}\end{array}\;\mbox{or}\;\begin{array}[]{c}\underset{(1\leq
a\leq
m)}{u_{3}^{a}=\displaystyle\frac{3u_{1}^{1}u_{2}^{1}u_{2}^{a}}{1+(u^{1}_{1})^{2}}}\\\
\end{array}\\\ \hline\cr\geq
3&\mathbb{A}_{2}^{\operatorname{tr}}&\mathfrak{M}-m-1&\begin{array}[]{c}\underset{(1\leq
a\leq
m)}{u_{n+1}^{a}=\displaystyle\frac{(n+1)u_{n}^{1}u_{n}^{a}}{nu^{1}_{n-1}}}\\\
\end{array}\\\ \hline\cr\geq
2&\mathbb{A}_{2}^{\operatorname{tf}}&\mathfrak{M}-2m+1+\delta_{2}^{n}&\begin{array}[]{c}\underset{(1\leq
a\leq m)}{u_{n+1}^{a}=(u_{n}^{2})^{2}\delta_{1}^{a}}\\\ \end{array}\\\
\hline\cr\end{array}$

(Recall $\mathfrak{M}=m^{2}+(n+1)m+3$ from (1.3).)

Table 1. Classification over $\mathbb{R}$ of submaximally symmetric vector
ODEs (1.2) of C-class of order $n+1\geq 3$

Our method for proving Theorems 1.1 and 1.2 will rely on the Cartan-geometric
viewpoint for vector ODEs, and the associated computations will be efficiently
done using representation theory. This will require important refinements to
the existing structural results for vector ODEs of C-class stated in Table 2.
Such refinements constitute our final main result, which we now briefly
describe. In our non-parabolic ODE setting, the aforementioned algebraic Hodge
theory establishes a $G_{0}$-equivariant identification of
$H^{2}(\mathfrak{g}_{-},\mathfrak{g})$ with the subspace
$\ker\square\subset\bigwedge^{2}\mathfrak{g}_{-}^{*}\otimes\mathfrak{g}$ of
harmonic 2-cochains (see §2.3). Analogous to Kostant’s theorem [14], which is
fundamental in the study of parabolic geometries, we may seek harmonic
realizations of lowest weight vectors $\Phi_{\mathbb{U}}\in\mathbb{U}$ for
each irreducible C-class submodule $\mathbb{U}\subset\mathbb{E}\subsetneq
H^{2}(\mathfrak{g}_{-},\mathfrak{g})$. Our Theorem 3.1 establishes such
realizations (see Table 4). We anticipate that these structural results will
be important for future geometric studies of the C-class and vector ODEs in
general.

##  2\. Cartan geometries and vector ODEs of C-class

In this section, we briefly review the Cartan-geometric reformulation for
vector ODEs (1.2) of order $\geq 3$ modulo point transformations, and
summarize all relevant facts about vector ODEs of C-class.

###  2.1. ODE geometry and symmetry

We begin by summarizing [13, §2.1], which is based on [10], and refer the
reader to these articles for more details. The $(n+1)$-st order ODE (1.2)
defines a submanifold $\mathcal{E}=\\{\mathbf{u}_{n+1}=\mathbf{f}\\}$ of co-
dimension $m\geq 2$ in the space of $(n+1)$-jets of functions
$J^{n+1}(\mathbb{R},\mathbb{R}^{m})$ that is transverse to the projection
$\pi_{n}^{n+1}:J^{n+1}(\mathbb{R},\mathbb{R}^{m})\to
J^{n}(\mathbb{R},\mathbb{R}^{m})$. Let $C$ denote the Cartan distribution on
$J^{n+1}(\mathbb{R},\mathbb{R}^{m})$ with standard local coordinates
$(t,\mathbf{u}_{0},\mathbf{u}_{1},\ldots,\mathbf{u}_{n+1})$, where
$\mathbf{u}_{r}=(u_{r}^{1},\ldots,u_{r}^{m})$. Then $C$ is given by

(2.1) $\displaystyle
C=\operatorname{span}\left\\{\partial_{t}+\mathbf{u}_{1}\partial_{\mathbf{u}_{0}}+\ldots+\mathbf{u}_{n+1}\partial_{\mathbf{u}_{n}},\quad\partial_{\mathbf{u}_{n+1}}\right\\},$

where
$\mathbf{u}_{i}\partial_{\mathbf{u}_{j}}:=\sum_{a=1}^{m}u_{i}^{a}\partial_{u_{j}^{a}}$
and $\partial_{\mathbf{u}_{r}}$ refers to
$\partial_{u_{r}^{1}},\ldots,\partial_{u_{r}^{m}}$. We also consider the
restriction of $C$ to $\mathcal{E}$ and abuse notation by also referring to
this distribution as $C$.

We will consider ODEs (1.2) up to point transformations. These are
diffeomorphisms $\Phi:J^{n+1}(\mathbb{R},\mathbb{R}^{m})\to
J^{n+1}(\mathbb{R},\mathbb{R}^{m})$ that preserve $C$, i.e. $d\Phi(C)=C$. By
the Lie–Bäcklund theorem, since $m\geq 2$, such transformations are the
prolongations of diffeomorphisms on
$J^{0}(\mathbb{R},\mathbb{R}^{m})\cong\mathbb{R}\times\mathbb{R}^{m}$.
Infinitesimally, a point vector field is a vector field
$\xi\in\mathfrak{X}(J^{n+1}(\mathbb{R},\mathbb{R}^{m}))$ whose flow is a point
transformation. Equivalently, $\mathcal{L}_{\xi}C\subset C$, where
$\mathcal{L}_{\xi}$ is the Lie derivative with respect to $\xi$. A point
symmetry of (1.2) is a point vector field that is tangent to $\mathcal{E}$.

The (point) geometry of $\mathcal{E}$ is encoded by a pair $(E,V)$ of
completely integrable sub-distributions of $C$ on $\mathcal{E}$:

(2.2) $\displaystyle
E=\operatorname{span}\left\\{\frac{d}{dt}:=\partial_{t}+\mathbf{u}_{1}\partial_{\mathbf{u}_{0}}+\cdots+\mathbf{u}_{n}\partial_{\mathbf{u}_{n-1}}+\mathbf{f}\partial_{\mathbf{u}_{n}}\right\\},\quad
V=\operatorname{span}\left\\{\partial_{\mathbf{u}_{n}}\right\\}.$

(Note that integral curves of $E$ are lifts of solution curves to (1.2).)
Moreover, the distribution $D:=E\oplus V\subset T\mathcal{E}$ is bracket-
generating, and its weak-derived flag defines the following filtration on
$T\mathcal{E}$:

(2.3) $\displaystyle T\mathcal{E}=D^{-n-1}\supset\cdots\supset D^{-2}\supset
D^{-1}:=D,$

where $D^{-i-1}:=D^{-i}+[D^{-i},D^{-1}]$ for $i>0$. Since
$[\Gamma(D^{j}),\Gamma(D^{k})]\subset\Gamma(D^{j+k})$, then the pair
$(\mathcal{E},\\{D^{i}\\})$ forms a filtered manifold. As we will describe
below, this leads to the formulation of an ODE (1.2) as a filtered
$G_{0}$-structure [3, §2.1].

Letting $T^{i}\mathcal{E}:=D^{i}\subset T\mathcal{E}$ for $-n-1\leq i\leq-1$
and $T^{0}\mathcal{E}:=0$, we define
$\operatorname{gr}(T\mathcal{E}):=\bigoplus^{-1}_{i=-n-1}\operatorname{gr}_{i}(T\mathcal{E})$
where
$\operatorname{gr}_{i}(T\mathcal{E}):=T^{i}\mathcal{E}/T^{i+1}\mathcal{E}$.
Let $\operatorname{gr}_{i}(T_{x}\mathcal{E})$ denote the fiber of
$\operatorname{gr}_{i}(T\mathcal{E})$ at $x\in\mathcal{E}$, i.e.
$\mathfrak{m}_{i}(x):=\operatorname{gr}_{i}(T_{x}\mathcal{E})=T^{i}_{x}\mathcal{E}/T^{i+1}_{x}\mathcal{E}$.
Then
$\mathfrak{m}(x):=\operatorname{gr}(T_{x}\mathcal{E})=\bigoplus^{-1}_{i=-n-1}\mathfrak{m}_{i}(x)$
is a nilpotent graded Lie algebra (NGLA) under the (Levi) bracket induced by
Lie bracket of vector fields. It is called the symbol algebra at $x$. Since
the symbol algebras at all points are isomorphic, then we let $\mathfrak{m}$
denote a fixed NGLA with $\mathfrak{m}\cong\mathfrak{m}(x),\,\forall
x\in\mathcal{E}$, and we say that $(\mathcal{E},\\{D^{i}\\})$ is regular of
type $\mathfrak{m}$.

Let
$\operatorname{Aut}_{\operatorname{gr}}(\mathfrak{m})\leq\operatorname{GL}(\mathfrak{m})$
be the subgroup that preserves the grading of $\mathfrak{m}$. Since
$\mathfrak{m}$ is generated by $\mathfrak{m}_{-1}$, then we have
$\operatorname{Aut}_{\operatorname{gr}}(\mathfrak{m})\hookrightarrow\operatorname{GL}(\mathfrak{m}_{-1})$.
For $x\in\mathcal{E}$, we let $F_{\operatorname{gr}}(x)$ denote the set of all
NGLA isomorphisms $\mathfrak{m}\to\mathfrak{m}(x)$. Then
$F_{\operatorname{gr}}(\mathcal{E}):=\bigcup_{x\in\mathcal{E}}F_{\operatorname{gr}}(x)$
defines a principal fiber bundle
$F_{\operatorname{gr}}(\mathcal{E})\to\mathcal{E}$ with structure group
$\operatorname{Aut}_{\operatorname{gr}}(\mathfrak{m})$, cf. [3, Prop 2.1]. The
splitting of $D$ implies a splitting of $\mathfrak{m}_{-1}$, and restricting
to the subgroup
$G_{0}\leq\operatorname{Aut}_{\operatorname{gr}}(\mathfrak{m})$ that preserves
the splitting yields a principal subbundle $\mathcal{G}_{0}\to\mathcal{E}$
with reduced structure group
$G_{0}\cong\mathbb{R}^{\times}\times\operatorname{GL}_{m}$. By [3, Defn 2.2],
an ODE (1.2) defines a filtered $G_{0}$-structure.

###  2.2. Structure underlying the trivial ODE

Let $n,m\geq 2$. The trivial ODE $\mathbf{u}_{n+1}=\mathbf{0}$ has point
symmetry Lie algebra $\mathfrak{g}$ (see for example [4, §2.2] for explicit
symmetry vector fields) with abstract structure given by

(2.4) $\displaystyle\mathfrak{g}\cong\mathfrak{q}\ltimes
V,\qquad\mathfrak{q}:=\mathfrak{sl}_{2}\times\mathfrak{gl}(W),\qquad
V:=\mathbb{V}_{n}\otimes W,\qquad W:=\mathbb{R}^{m},$

where $\mathbb{V}_{n}$ is the unique (up to isomorphism)
$\mathfrak{sl}_{2}$-irrep of dimension $n+1$, and $W$ is the standard rep of
$\mathfrak{gl}(W)$. Here, $V$ is taken to be an abelian subalgebra.

We now fix a basis for $\mathfrak{g}$. Let $\\{w_{a}\\}_{a=1}^{m}$ be the
standard basis for $W$, and let $e_{a}{}^{b}$ be the $m\times m$ matrix such
that $e_{a}{}^{b}w_{c}=\delta_{c}{}^{b}w_{a}$, so that
$\\{e_{a}{}^{b}\\}_{a,b=1}^{m}$ spans $\mathfrak{gl}(W)$. Letting $\\{x,y\\}$
be the standard basis for $\mathbb{R}^{2}$, we identify $\mathbb{V}_{n}\cong
S^{n}\mathbb{R}^{2}$. We obtain bases $\\{E_{i}\\}_{i=0}^{n}$ on
$\mathbb{V}_{n}$ and $\\{E_{i,a}:0\leq i\leq n,\,1\leq a\leq m\\}$ on $V$ via

(2.5) $\displaystyle E_{i}:=\dfrac{x^{i}y^{n-i}}{(n-i)!},\qquad
E_{i,a}:=E_{i}\otimes w_{a}.$

(For convenience, we define $E_{i}=0$ for $i<0$ or $i>n$. We also caution that
our $E_{i}$ corresponds to $E_{n-i}$ in [13, §2.1.2].) We complete our bases
of $V$ and $\mathfrak{gl}(W)$ to a basis of $\mathfrak{g}$ by introducing the
standard $\mathfrak{sl}_{2}$-triple

(2.6)
$\displaystyle\mathsf{X}:=x\partial_{y},\qquad\mathsf{H}:=x\partial_{x}-y\partial_{y},\qquad\mathsf{Y}:=y\partial_{x}.$

Note that $\mathfrak{sl}_{2}$ commutes with $\mathfrak{gl}(W)$, and the
$\mathfrak{sl}_{2}$-actions on $\mathbb{V}_{n}$ and $V$ are naturally induced,
e.g.

(2.7)
$\displaystyle[\mathsf{X},E_{i}]=E_{i+1},\qquad[\mathsf{H},E_{i}]=(2i-n)E_{i},\qquad[\mathsf{Y},E_{i}]=i(n+1-i)E_{i-1}.$

In particular, $E_{i}$ and $E_{i,a}$ are weight vectors for the
$\mathfrak{sl}_{2}$-action, i.e. eigenvectors with respect to $\mathsf{H}$.

Now endow $\mathfrak{g}$ with a bi-grading as in [13, §3.1]. Letting
$\operatorname{id}_{m}:=\sum_{a=1}^{m}e_{a}{}^{a}$, define
$\mathsf{Z}_{1},\mathsf{Z}_{2}\in\mathfrak{g}$ by

(2.8)
$\displaystyle\mathsf{Z}_{1}:=-\frac{1}{2}(\mathsf{H}+n\operatorname{id}_{m}),\qquad\mathsf{Z}_{2}:=-\operatorname{id}_{m}.$

Then $\mathfrak{g}$ decomposes into the joint eigenspaces of
$\operatorname{ad}_{\mathsf{Z}_{1}}$ and $\operatorname{ad}_{\mathsf{Z}_{2}}$.
We write

(2.9)
$\displaystyle\mathfrak{g}_{s,t}:=\\{x\in\mathfrak{g}\,:\,\mathsf{Z}_{1}\cdot
x=sx,\,\,\mathsf{Z}_{2}\cdot x=tx\\},$

and refer to $s$ and $t$ as the $\mathsf{Z}_{1}$-degree and
$\mathsf{Z}_{2}$-degree of $x$, respectively. The ordered pair
$(s,t)\in\mathbb{Z}\times\mathbb{Z}$ is the bi-grade of $x$. It is helpful to
picture $\mathfrak{g}$ as in Figure 1.

$\cdots$$(-1,0)$$(0,0)$$(1,0)$$(0,-1)$$(-1,-1)$$(-n+1,-1)$$(-n,-1)$$\mathsf{X}$$\mathsf{H},e_{a}{}^{b}$$\mathsf{Y}$$E_{n,a}$$E_{n-1,a}$$E_{1,a}$$E_{0,a}$
Figure 1. Bi-grading on $\mathfrak{g}$

Defining the grading element $\mathsf{Z}\in\mathfrak{z}(\mathfrak{g}_{0,0})$,
we similarly induce the structure of a $\mathbb{Z}$-grading on $\mathfrak{g}$
via

(2.10)
$\displaystyle\mathsf{Z}:=\mathsf{Z}_{1}+\mathsf{Z}_{2}=-\frac{1}{2}\left(\mathsf{H}+(n+2)\operatorname{id}_{m}\right).$

(In a given representation, $\mathsf{Z}$-eigenvalues will also be referred to
as degrees.) Then we have the decomposition
$\mathfrak{g}=\mathfrak{g}_{-n-1}\oplus\ldots\oplus\mathfrak{g}_{1}$, where

(2.11)
$\displaystyle\begin{split}\mathfrak{g}_{1}&:=\mathfrak{g}_{1,0}=\mathbb{R}\mathsf{Y},\\\
\mathfrak{g}_{0}&:=\mathfrak{g}_{0,0}=\mathbb{R}\mathsf{H}\oplus\mathfrak{gl}_{m},\\\
\mathfrak{g}_{-1}&:=\mathfrak{g}_{-1,0}\oplus\mathfrak{g}_{0,-1}=\mathbb{R}\mathsf{X}\oplus(\mathbb{R}E_{0}\otimes
W),\\\ \mathfrak{g}_{i}&:=\mathfrak{g}_{i+1,-1}=\mathbb{R}E_{i+1}\otimes
W,\qquad i=-2,\ldots,-n-1.\end{split}$

We note that
$\mathfrak{g}_{-}:=\mathfrak{g}_{-n-1}\oplus\ldots\oplus\mathfrak{g}_{-1}\subset\mathfrak{g}$
is generated by $\mathfrak{g}_{-1}$.

We also endow $\mathfrak{g}$ with the canonical filtration
$\mathfrak{g}^{i}:=\sum_{j\geq i}\mathfrak{g}_{j}$, which turns $\mathfrak{g}$
into a filtered Lie algebra. Its associated graded
$\operatorname{gr}(\mathfrak{g}):=\bigoplus_{k\in\mathbb{Z}}\operatorname{gr}_{k}(\mathfrak{g})$,
where
$\operatorname{gr}_{k}(\mathfrak{g})=\mathfrak{g}^{k}/\mathfrak{g}^{k+1}$, is
isomorphic to $\mathfrak{g}$ as graded Lie algebras. Using the isomorphism, we
let $\operatorname{gr}_{k}:\mathfrak{g}^{k}\to\mathfrak{g}_{k}$ denote the
leading part. Explicitly, if $x\in\mathfrak{g}^{k}$ with
$x=x_{k}+x_{k+1}+\ldots$, where $x_{j}\in\mathfrak{g}_{j}$, then
$\operatorname{gr}_{k}(x):=x_{k}$. The following notations will be convenient:

(2.12)
$\displaystyle\mathfrak{p}:=\mathfrak{g}^{0}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1},\qquad\mathfrak{p}_{+}:=\mathfrak{g}^{1}=\mathfrak{g}_{1}.$

At the group level, let

(2.13) $\displaystyle
G:=(\operatorname{SL}_{2}\times\operatorname{GL}_{m})\ltimes V,\qquad
P:=\operatorname{ST}_{2}\times\operatorname{GL}_{m},\qquad G_{0}:=\\{g\in
P:\operatorname{Ad}_{g}(\mathfrak{g}_{0})\subset\mathfrak{g}_{0}\\},$

where $\operatorname{ST}_{2}\subset\operatorname{SL}_{2}$ is the subgroup of
lower triangular matrices. (Note that $G_{0}$ is isomorphic to that given in
§2.1.) We also let $P_{+}\subset P$ denote the connected Lie subgroup
corresponding to $\mathfrak{p}_{+}\subset\mathfrak{p}$. We remark that the
canonical filtration on $\mathfrak{g}$ is $P$-invariant.

###  2.3. Cartan geometries associated to ODE

Fix $G$, $P$ and $G_{0}$ as above. Recall also from §2.1 that all vector ODEs
(1.2) can be formulated as filtered $G_{0}$-structures. Importantly, there is
an equivalence of categories between filtered $G_{0}$-structures on
$\mathcal{E}$ (which is a wider category than that arising from ODE – see
below) and regular, normal Cartan geometries
$(\mathcal{G}\to\mathcal{E},\omega)$ of type $(G,P)$ [4, 10]. A Cartan
geometry consists of a (right) principal $P$-bundle
$\mathcal{G}\to\mathcal{E}$ endowed with a Cartan connection $\omega$, i.e.
$\omega\in\Omega^{1}(\mathcal{G},\mathfrak{g})$ is a $\mathfrak{g}$-valued
1-form on $\mathcal{G}$ such that:

  1. (a)

For any $u\in\mathcal{G}$, $\omega_{u}:T_{u}\mathcal{G}\to\mathfrak{g}$ is a
linear isomorphism;

  2. (b)

$R_{g}^{\ast}\omega=\operatorname{Ad}_{g^{-1}}\circ\,\omega$ for any $g\in P$,
i.e. $\omega$ is $P$-equivariant;

  3. (c)

$\omega(\zeta_{A})=A$, where $A\in\mathfrak{p}$, where $\zeta_{A}$ is the
fundamental vertical vector field defined by
$\zeta_{A}(u):=\frac{d}{dt}{\big{|}}_{t=0}\,u\cdot\exp(tA)$.

The curvature $K\in\Omega^{2}(\mathcal{G},\mathfrak{g})$ of the geometry is
given by $K(\xi,\eta)=d\omega(\xi,\eta)+[\omega(\xi),\omega(\eta)]$, which is
$P$-equivariant and horizontal, i.e. $K(\zeta_{A},\cdot)=0,\,\forall
A\in\mathfrak{p}$. Consequently, $K$ is determined by the $P$-equivariant
curvature function
$\kappa:\mathcal{G}\to\bigwedge^{2}(\nicefrac{{\mathfrak{g}}}{{\mathfrak{p}}})^{\ast}\otimes\mathfrak{g}$,
defined by
$\kappa(u)(A,B)=K(\omega^{-1}(A),\omega^{-1}(B))(u),\,\,A,B\in\mathfrak{g}$.
Letting $\omega_{G}$ be the Maurer–Cartan form on $G$, the Klein geometry
$(G\to\nicefrac{{G}}{{P}},\omega_{G})$ satisfies $K\equiv 0$ (Maurer–Cartan
equation), and is the flat model for all Cartan geometries of type $(G,P)$.

In terms of the canonical filtration $\\{\mathfrak{g}^{i}\\}$ on
$\mathfrak{g}$ from §2.2, $\omega$ is said to be regular if
$\kappa(\mathfrak{g}^{i},\mathfrak{g}^{j})\subset\mathfrak{g}^{i+j+1}\,\,\forall
i,j$. Importantly, it is known that for all filtered $G_{0}$-structures
arising from ODE, the corresponding Cartan geometry has $\kappa$ satisfying
the strong regularity condition [4, Rem 2.3]:

(2.14)
$\displaystyle\kappa(\mathfrak{g}^{i},\mathfrak{g}^{j})\subset\mathfrak{g}^{i+j+1}\cap\mathfrak{g}^{\min(i,j)-1},\quad\forall
i,j.$

To define normality, we first fix an inner product $\langle\cdot,\cdot\rangle$
on $\mathfrak{g}$ in terms of the basis introduced in §2.2:

######  Definition 2.1.

Let $\langle\cdot,\cdot\rangle$ be an inner product on $\mathfrak{g}$ such
that $\\{\mathsf{X},\mathsf{H},\mathsf{Y},e_{a}{}^{b},E_{i,a}\\}$ is an
orthogonal basis for $\mathfrak{g}$ with squared lengths of basis elements
given below:

(2.15)
$\displaystyle\langle\mathsf{X},\mathsf{X}\rangle=\langle\mathsf{Y},\mathsf{Y}\rangle=1,\quad\langle\mathsf{H},\mathsf{H}\rangle=2,\quad\langle
e_{a}{}^{b},e_{a}{}^{b}\rangle=1,\quad\langle
E_{i,a},E_{i,a}\rangle=\frac{i!}{(n-i)!}.$

Then $\forall A,B\in\mathfrak{q}=\mathfrak{sl}_{2}\times\mathfrak{gl}_{m}$ and
$\forall u,v\in V$, we have $\langle A,B\rangle=\operatorname{tr}(A^{T}B)$ and
$\langle Au,v\rangle=\langle u,A^{T}v\rangle$.

Consider
$C^{k}(\mathfrak{g},\mathfrak{g}):=\bigwedge^{k}\mathfrak{g}^{\ast}\otimes\mathfrak{g}$
equipped with the induced canonical filtration from $\mathfrak{g}$ and let
$\partial_{\mathfrak{g}}$ be the standard differential of the complex for
computing Lie algebra cohomology groups $H^{k}(\mathfrak{g},\mathfrak{g})$.
Then, define the codifferential
$\partial^{\ast}:C^{k}(\mathfrak{g},\mathfrak{g})\to
C^{k-1}(\mathfrak{g},\mathfrak{g})$ to be the adjoint of
$\partial_{\mathfrak{g}}$ with respect to the induced inner product from
$\mathfrak{g}$, i.e. for each $k$ we have
$\langle\partial_{\mathfrak{g}}\phi,\psi\rangle=\langle\phi,\partial^{\ast}\psi\rangle$
for all $\phi\in C^{k-1}(\mathfrak{g},\mathfrak{g})$ and $\psi\in
C^{k}(\mathfrak{g},\mathfrak{g})$. By [4, Lemma 3.2], the codifferential
descends to a $P$-equivariant map
$\partial^{\ast}:\bigwedge^{k}(\nicefrac{{\mathfrak{g}}}{{\mathfrak{p}}})^{\ast}\otimes\mathfrak{g}\to\bigwedge^{k-1}(\nicefrac{{\mathfrak{g}}}{{\mathfrak{p}}})^{\ast}\otimes\mathfrak{g}$.
A Cartan connection $\omega$ has curvature function $\kappa$ valued in
$\bigwedge^{2}(\nicefrac{{\mathfrak{g}}}{{\mathfrak{p}}})^{*}\otimes\mathfrak{g}$,
and $\omega$ is said to be normal if $\partial^{\ast}\kappa=0$. In this
article, we will always work with Cartan geometries of type $(G,P)$ that are
normal and strongly regular.

Since $(\partial^{\ast})^{2}=0$, then the (normal) curvature $\kappa$
quotients to a $P$-equivariant function
$\kappa_{H}:\mathcal{G}\to\frac{\ker\partial^{\ast}}{\operatorname{im}\partial^{\ast}}$
called the harmonic curvature. By regularity, $\kappa_{H}$ is valued in the
filtrand of positive degree of the $P$-module
$\frac{\ker\partial^{\ast}}{\operatorname{im}\partial^{\ast}}$, which by [4,
Corollary 3.8] is completely reducible, i.e. $P_{+}$ acts on it trivially, and
therefore only the $G_{0}$-action is relevant. It is well-known (see Theorem
A.3 and references therein) that $\kappa_{H}$ completely obstructs local
flatness, i.e. $\kappa_{H}\equiv 0\iff\kappa=0$.

Identify
$\bigwedge^{k}(\nicefrac{{\mathfrak{g}}}{{\mathfrak{p}}})^{*}\otimes\mathfrak{g}\cong\bigwedge^{k}\mathfrak{g}_{-}^{*}\otimes\mathfrak{g}$
as $G_{0}$-modules, and recall from §2.2 that
$\mathfrak{g}\cong\mathfrak{q}\ltimes V$. Given $\phi\in
C^{k}(\mathfrak{g}_{-},\mathfrak{g}):=\bigwedge^{k}\mathfrak{g}_{-}^{*}\otimes\mathfrak{g}$,
then we have $\phi=\mathsf{X}^{*}\wedge\phi_{1}+\phi_{2}$, for $\phi_{1}\in
C^{k-1}(V,\mathfrak{g})$ and $\phi_{2}\in C^{k}(V,\mathfrak{g})$, and where
$\mathsf{X}^{*}$ is dual to $\mathsf{X}$. Denoting
$\phi:=\begin{pmatrix}\phi_{1}\\\ \phi_{2}\end{pmatrix}$, then $\partial\phi$
is given by [4, Lem 3.4]:

(2.16) $\displaystyle\partial\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=\begin{pmatrix}-\partial_{V}\phi_{1}+\mathsf{X}\cdot\phi_{2}\\\
\partial_{V}\phi_{2}\end{pmatrix},$

where

(2.17)
$\displaystyle\partial_{V}\phi_{2}(x_{0},\ldots,x_{k})=\sum_{i=0}^{k}(-1)^{i}x_{i}\cdot\phi_{2}(x_{0},\ldots,\widehat{x}_{i},\ldots,x_{k}),$

for $x_{0},\ldots,x_{k}\in V$, and letting $\widehat{x}_{i}$ denote omission
of $x_{i}$. A direct consequence of (2.16) is:

######  Lemma 2.2.

Let $\phi\in\bigwedge^{k}V^{*}\otimes\mathfrak{g}$. Then $\partial\phi=0$ if
and only if $\mathsf{X}\cdot\phi=0$ and $\partial_{V}\phi=0$. Moreover, if in
fact $\phi\in\bigwedge^{k}V^{*}\otimes V$, then $\partial\phi=0$ if and only
if $\mathsf{X}\cdot\phi=0$.

Defining
$\square:=\partial\circ\partial^{\ast}+\partial^{\ast}\circ\partial:\bigwedge^{k}\mathfrak{g}_{-}^{*}\otimes\mathfrak{g}\to\bigwedge^{k}\mathfrak{g}_{-}^{*}\otimes\mathfrak{g}$,
we then have the following $G_{0}$-isomorphisms,

(2.18)
$\displaystyle\bigwedge{}^{\\!\\!k}{\mathfrak{g}_{-}^{\ast}\otimes\mathfrak{g}}\cong\hbox
to0.0pt{$\overbrace{\phantom{\,\operatorname{im}\partial^{\ast}\oplus\ker\square}}^{\ker\partial^{\ast}}$\hss}\operatorname{im}\partial^{\ast}\oplus\underbrace{\ker\square\oplus\operatorname{im}\partial}_{\ker\partial},\qquad\ker\square\cong\frac{\ker\partial^{\ast}}{\operatorname{im}\partial^{\ast}}\cong\frac{\ker\partial}{\operatorname{im}\partial}=:H^{k}(\mathfrak{g}_{-},\mathfrak{g}).$

Consequently, for a regular, normal Cartan geometry, the codomain of
$\kappa_{H}$ can be identified with the subspace
$H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})\subset
H^{2}(\mathfrak{g}_{-},\mathfrak{g})$ on which the grading element
$\mathsf{Z}=\mathsf{Z}_{1}+\mathsf{Z}_{2}$ acts with positive eigenvalues.
However, it should be emphasized that only part of
$H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})$ is in fact realizable for
geometries associated to ODE [18, 11]. Correspondingly, we define:

######  Definition 2.3.

The effective part $\mathbb{E}\subsetneq
H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})$ is the minimal $G_{0}$-module in
which $\kappa_{H}$ is valued, for any (strongly) regular, normal Cartan
geometry of type $(G,P)$ associated to an ODE (1.2) (for fixed $(n,m)$).

###  2.4. Vector ODEs of C-class

We will focus on ODEs (1.2) of C-class, which have been characterized in [4]
using curvatures $\kappa$ of corresponding canonical Cartan connections
$\omega$ described above. We define [4, Defn 2.4]:

######  Definition 2.4.

An ODE (1.2) is said to be of C-class if the curvature $\kappa$ of the
corresponding strongly regular, normal Cartan geometry satisfies
$\kappa(\mathsf{X},\cdot)=0$, where $\mathsf{X}\in\mathfrak{g}_{-1}$ was
defined in §2.2.

######  Remark 2.5.

Recall from §2.2 that $\mathfrak{g}\cong\mathfrak{q}\ltimes V$. We remark that
for a Cartan geometry corresponding to an ODE of C-class, we can identify
$\kappa\in\bigwedge^{2}(\mathfrak{g}/\mathfrak{q})^{*}\otimes\mathfrak{g}\cong\bigwedge^{2}V^{\ast}\otimes\mathfrak{g}$.

As shown in [4], the notion of C-class can be concretely reformulated in terms
of fundamental invariants for vector ODEs (1.2) of order $\geq 3$ described
below, which comprise the harmonic curvature of the geometry. We then have the
following characterization of the C-class given in [4, Thms 4.1 & 4.2]:

######  Theorem 2.6.

A vector ODE (1.2) of order $\geq 3$ is of C-class if and only if all of its
generalized Wilczynski invariants vanish.

For concreteness, we now explicitly describe the fundamental invariants for
vector ODEs (1.2) of order $n+1\geq 3$ consisting of generalized Wilczynski
invariants $\mathcal{W}_{r}$ [8] and C-class invariants [18, 19, 11]:

  * •

Consider a linear vector ODE of order $n+1$:

(2.19)
$\displaystyle\mathbf{u}_{n+1}+P_{n}(t)\mathbf{u}_{n}+\ldots+P_{1}(t)\mathbf{u}_{1}+P_{0}(t)\mathbf{u}=0,$

where $P_{j}(t)$ is an $\operatorname{End}(\mathbb{R}^{m})$-valued function.
Using the invertible transformations
$(t,\mathbf{u})\mapsto(f(t),h(t)\mathbf{u})$ where
$f:\mathbb{R}\to\mathbb{R}^{\times}$ and
$h:\mathbb{R}\to\operatorname{GL}(m)$, which preserve the form of equation
(2.19), we may normalize to $P_{n}=0$ and $\operatorname{tr}(P_{n-1})=0$, i.e.
Laguerre–Forsyth canonical form. Then

(2.20)
$\displaystyle\Theta_{r}=\sum_{k=1}^{r-1}(-1)^{k+1}\dfrac{(2r-k-1)!(n-r+k)!}{(r-k)!(k-1)!}P_{n-r+k}^{(k-1)},\quad
r=2,\ldots,n+1,$

are fundamental invariants found by Se-ashi [22], and $r$ is the degree of the
invariant. For (1.2), the generalized Wilczynski invariants $\mathcal{W}_{r}$
(for $r=2,\ldots,n+1$) are defined as $\Theta_{r}$ above evaluated at its
linearization along a solution $\mathbf{u}$. Formally, $\mathcal{W}_{r}$ are
obtained from (2.19) by replacing $P_{r}(t)$ by the matrices
$-\left(\frac{\partial f^{a}}{\partial u_{r}^{b}}\right)$ and the usual
derivative by the total derivative $\frac{d}{dt}$ given in (2.2). Moreover,
$\mathcal{W}_{r}$ do not depend on the choice of solution $\mathbf{u}$, and
are therefore contact invariants.

  * •

C-class invariants are the following:

(2.21) $\displaystyle\begin{split}n\geq
2:\,\,{(\mathcal{A}_{2}^{\operatorname{tf}})}^{a}_{bc}&=\operatorname{tf}\left(\frac{\partial^{2}{f}^{a}}{\partial{u_{n}^{b}}\,\partial{u_{n}^{c}}}\right),\\\
n\geq
3:\,\,{(\mathcal{A}_{2}^{\operatorname{tr}})}^{a}_{bc}&=\operatorname{tr}\left(\frac{\partial^{2}{f}^{a}}{\partial{u_{n}^{b}}\,\partial{u_{n}^{c}}}\right),\\\
n=2:\,\,\,(\mathcal{B}_{4})_{bc}&=-\frac{\partial H_{c}^{-1}}{\partial
u_{1}^{b}}+\frac{\partial}{\partial u_{2}^{b}}\frac{\partial}{\partial
u_{2}^{c}}{H^{t}}-\frac{\partial}{\partial
u_{2}^{c}}\frac{d}{dt}{H^{-1}_{b}}-\frac{\partial}{\partial
u_{2}^{c}}\left(\sum_{a=1}^{m}H_{a}^{-1}\frac{\partial f^{a}}{\partial
u_{2}^{b}}\right)+2H^{-1}_{b}H^{-1}_{c},\end{split}$

where

(2.22) $\displaystyle
H_{b}^{-1}=\frac{1}{6(m+1)}\sum_{a=1}^{m}\frac{\partial^{2}{f}^{a}}{\partial{u_{2}^{a}}\,\partial{u_{2}^{b}}},\qquad
H^{t}=-\frac{1}{4m}\sum_{a=1}^{m}\left(\frac{\partial f^{a}}{\partial
u_{1}^{a}}-\frac{d}{dt}\frac{\partial f^{a}}{\partial
u_{2}^{a}}+\frac{1}{3}\sum_{c=1}^{m}\frac{\partial f^{a}}{\partial
u_{2}^{c}}\frac{\partial f^{c}}{\partial u_{2}^{a}}\right).$

###  2.5. C-class modules

The above fundamental invariants correspond to $G_{0}$-irreducible submodules
in the effective part $\mathbb{E}\subsetneq
H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})$ (Definition 2.3), which we now
describe. Recall that
$\mathfrak{g}_{0}\cong\operatorname{span}\\{\mathsf{Z}_{1},\mathsf{Z}_{2}\\}\oplus\mathfrak{sl}(W)$,
and we have the induced action of $\mathsf{Z}_{1}$ and $\mathsf{Z}_{2}$ from
(2.8) on $H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})$, and therefore on
$\mathbb{E}$. Note that $\mathsf{Z}_{2}$ acts with degrees $0,1$ or $2$. We
define:

######  Definition 2.7.

A $G_{0}$-submodule $\mathbb{U}\subset\mathbb{E}\subsetneq
H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})$ on which $\mathsf{Z}_{2}$ acts with
positive degree(s) is called a C-class module, and we let
$\mathbb{E}_{C}\subsetneq\mathbb{E}$ denote the direct sum of all irreducible
C-class modules. On the other hand, if $\mathsf{Z}_{2}$ acts on $\mathbb{U}$
with zero degree, we refer to $\mathbb{U}$ as a Wilczynski module.

Any $\mathfrak{g}_{0}$-irrep $\mathbb{U}\subset\mathbb{E}$ is determined by
its bi-grade and its lowest weight $\lambda$ with respect to
$\mathfrak{sl}(W)\cong\mathfrak{sl}_{m}$. Such $\lambda$ can be expressed in
terms of the fundamental weights $\lambda_{1},\ldots,\lambda_{m-1}$ of
$\mathfrak{sl}_{m}$ with respect to the Cartan subalgebra $\mathfrak{h}$
consisting of diagonal matrices in $\mathfrak{sl}_{m}$, and the standard
choice of $m-1$ simple roots. Letting
$h=\operatorname{diag}(h_{1},\ldots,h_{m})\in\mathfrak{h}$ and
$\epsilon_{a}:\mathfrak{h}\to\mathbb{R}$ the linear functional
$\epsilon_{a}(h)=h_{a}$, we then have $\epsilon_{1}+...+\epsilon_{m}=0$ and
$\lambda_{i}=\epsilon_{1}+\ldots+\epsilon_{i}$ for $1\leq i\leq m-1$.

Table 2 contains a summary of results for $\mathbb{E}_{C}\subsetneq\mathbb{E}$
for ODEs (1.2), due to Medvedev [18, 19] for order $3$ and Doubrov–Medvedev
[11] for order $\geq 4$. Using the $G_{0}$-isomorphisms (2.18), we identify
each irreducible C-class module $\mathbb{U}\subset\mathbb{E}_{C}$ from Table 2
with the corresponding module in $\ker\square\subset
C^{2}(\mathfrak{g}_{-},\mathfrak{g})$ consisting of harmonic 2-cochains
satisfying the strong regularity condition (2.14). (A further condition is
formulated in §2.6 below.) Adopting the same notation from [13, Table 6], we
let $\mathbb{A}_{2}$ and $\mathbb{B}_{4}$ denote the C-class submodules with
bi-grades $(1,1)$ and $(2,2)$ respectively. From the respective
$\mathsf{Z}_{2}$-degrees, and since
$\kappa\in\bigwedge^{2}V^{*}\otimes\mathfrak{g}$ for C-class ODE, then we
deduce that we may identify

(2.23) $\displaystyle\mathbb{A}_{2}\subset\bigwedge^{2}V^{*}\otimes
V,\qquad\mathbb{B}_{4}\subset\bigwedge^{2}V^{*}\otimes\mathfrak{q}.$

Since $\mathbb{A}_{2}$ is not irreducible, we decompose it into (irreducible)
trace and trace-free parts:
$\mathbb{A}_{2}=\mathbb{A}_{2}^{\operatorname{tr}}\oplus\mathbb{A}_{2}^{\operatorname{tf}}$.
(The C-class invariants
$\mathcal{B}_{4},\mathcal{A}_{2}^{\operatorname{tr}},\mathcal{A}_{2}^{\operatorname{tf}}$
from §2.4 are valued in the corresponding irreducible C-class modules
$\mathbb{B}_{4},\mathbb{A}_{2}^{\operatorname{tr}},\mathbb{A}_{2}^{\operatorname{tf}}$
respectively.)

$\begin{array}[]{|ccccc|}\hline\cr n&\mbox{Irreducible C-class
module}\,\mathbb{U}&\mbox{Bi-grade}&\mathfrak{sl}(W)\mbox{-module
structure}&\mathfrak{sl}(W)\mbox{-lowest weight }\lambda\\\ \hline\cr\hline\cr
2&\mathbb{B}_{4}&(2,2)&S^{2}W^{\ast}&-2\epsilon_{1}\\\ \geq
3&\mathbb{A}_{2}^{\operatorname{tr}}&(1,1)&W^{\ast}&-\epsilon_{1}\\\ \geq
2&\mathbb{A}_{2}^{\operatorname{tf}}&(1,1)&(S^{2}W^{\ast}\otimes
W)_{0}&\epsilon_{m}-2\epsilon_{1}\\\ \hline\cr\end{array}$ Table 2. C-class
modules in $\mathbb{E}_{C}\subsetneq\mathbb{E}\subsetneq
H^{2}_{+}(\mathfrak{g}_{-},\mathfrak{g})$ for vector ODEs of order $n+1\geq 3$

Since each $\mathbb{U}$ is a $\mathfrak{g}_{0}$-irrep, then up to scale
$\mathbb{U}$ contains a unique lowest weight vector $\Phi_{\mathbb{U}}$. Since
$\mathfrak{g}_{0}\cong\operatorname{span}\\{\mathsf{Z}_{1},\mathsf{Z}_{2}\\}\oplus\mathfrak{sl}(W)$,
then being “lowest” means that $\Phi_{\mathbb{U}}$ is annihilated by all
lowering operators, i.e. strictly lower triangular matrices, in
$\mathfrak{sl}(W)\cong\mathfrak{sl}_{m}$. From Table 2, we can give an
explicit description of the annihilators $\mathfrak{ann}(\Phi_{\mathbb{U}})$,
which will be needed later. Namely, if
$\tilde{\mathfrak{p}}\subset\mathfrak{sl}_{m}$ is the parabolic subalgebra
preserving $\Phi_{\mathbb{U}}$ up to scale, then
$\mathfrak{ann}(\Phi_{\mathbb{U}})\subset\operatorname{span}\\{\mathsf{Z}_{1},\mathsf{Z}_{2}\\}\oplus\tilde{\mathfrak{p}}$.
For $a\neq c$, if $e_{a}{}^{c}\in\tilde{\mathfrak{p}}$, then
$e_{a}{}^{c}\in\mathfrak{ann}(\Phi_{\mathbb{U}})$. It suffices to consider
linear combinations of $\mathsf{Z}_{1}$, $\mathsf{Z}_{2}$, and diagonal
elements $\mathfrak{h}\subset\tilde{\mathfrak{p}}$. If $\Phi_{\mathbb{U}}$ has
$\mathfrak{sl}_{m}$-weight $\lambda$ and $\mathsf{Z}_{2}$-degree $t$, then we
conclude that $\mathfrak{ann}(\Phi_{\mathbb{U}})$ is spanned by

(2.24) $\displaystyle\mathsf{Z}_{1}-\mathsf{Z}_{2},\qquad
h-\frac{\lambda(h)}{t}\mathsf{Z}_{2}\,\,\,(h\in\mathfrak{h}),\qquad
e_{a}{}^{c}\in\tilde{\mathfrak{p}}\,\,\,(a\neq c),$

where $\mathsf{Z}_{1}-\mathsf{Z}_{2}\in\mathfrak{ann}(\Phi_{\mathbb{U}})$
because of the bi-grading of $\mathbb{U}$. Applying (2.24) to $(\lambda,t)$
from Table 2, we obtain Table 3. Here,
$\tilde{\mathfrak{p}}_{1},\,\tilde{\mathfrak{p}}_{1,m-1}$ are the parabolic
subalgebras in $\mathfrak{sl}_{m}$ consisting of block lower triangular
matrices with diagonal blocks of sizes $1,m-1$ and $1,m-2,1$ respectively.

$\begin{array}[]{|cccl|}\hline\cr
n&\mathbb{U}&\operatorname{dim}\mathfrak{ann}(\Phi_{\mathbb{U}})&\lx@intercol\hfil\mbox{Generators
for
}\mathfrak{ann}(\Phi_{\mathbb{U}})\subset\mathfrak{g}_{0}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr
2&\mathbb{B}_{4}&\hbox{\multirowsetup$m^{2}-m+1$}&\hbox{\multirowsetup$\begin{array}[]{l}\mathsf{Z}_{1}-\mathsf{Z}_{2},\,\,e_{a}{}^{c}\in\tilde{\mathfrak{p}}_{1}\,\,(a\neq
c),\\\ e_{b}{}^{b}-e_{b+1}{}^{b+1}+\delta_{1}{}^{b}\mathsf{Z}_{2}\,\,(1\leq
b\leq m-1)\\\ \end{array}$}\\\ \geq 3&\mathbb{A}_{2}^{\operatorname{tr}}&&\\\
\hline\cr\geq
2&\mathbb{A}_{2}^{\operatorname{tf}}&m^{2}-2m+3&\begin{array}[]{l}\mathsf{Z}_{1}-\mathsf{Z}_{2},\,\,e_{a}{}^{c}\in\tilde{\mathfrak{p}}_{1,m-1}\,\,(a\neq
c),\\\
e_{b}{}^{b}-e_{b+1}{}^{b+1}+(2\delta_{1}{}^{b}+\delta_{m-1}{}^{b})\mathsf{Z}_{2}\,\,(1\leq
b\leq m-1)\\\ \end{array}\\\ \hline\cr\end{array}$
Table 3. $\mathfrak{ann}(\Phi_{\mathbb{U}})\subset\mathfrak{g}_{0}$ for
irreducible C-class modules $\mathbb{U}\subset\mathbb{E}$

###  2.6. The Doubrov–Medvedev condition

We will be able to precisely identify $\mathbb{A}_{2}$ with the help of an
additional linear condition formulated in [11, §3.1, Prop.4], and which we now
summarize. Consider the $\mathfrak{p}$-invariant subspace
$F=\operatorname{span}\\{E_{0},\ldots,E_{n-1}\\}\otimes W\subset V$, and
define
$\delta:\operatorname{Hom}(F,\mathbb{R}\mathsf{X})\to\operatorname{Hom}\left(\bigwedge^{2}F,V/F\right)$
by

(2.25) $\displaystyle(\delta B)(x,y)=(B(x)\cdot y-B(y)\cdot x)\mod
F,\quad\forall B\in\operatorname{Hom}(F,\mathbb{R}\mathsf{X}).$

We have the inclusion $\iota_{F}:F\to V$, which induces $V/F\cong W$ (as
$\mathfrak{p}$-modules) and natural quotient $\pi_{W}:V\to V/F$. Also induced
is the inclusion $\iota_{\bigwedge^{2}F}:\bigwedge^{2}F\to\bigwedge^{2}V$,
from which we define
$\vartheta:\operatorname{Hom}\left(\bigwedge^{2}V,V\right)\to\operatorname{Hom}\left(\bigwedge^{2}F,V/F\right)$
by $\vartheta=\pi_{W}\circ\iota_{\bigwedge^{2}F}^{*}$, i.e.

(2.26) $\displaystyle\vartheta(A)=\left.A\right|_{\bigwedge^{2}F}\mod F.$

From [11, §3.1, Prop 4] and Remark 2.5, we deduce that for a C-class ODE of
order $\geq 4$, the $\mathbb{A}_{2}$-component $\mathcal{A}_{2}$ of its
harmonic curvature $\kappa_{H}$ satisfies
$\vartheta(\mathcal{A}_{2})\in\operatorname{im}(\delta)$, which we refer to as
the Doubrov–Medvedev condition. Correspondingly, for $n\geq 3$ we formulate
the algebraic condition

(2.27) $\displaystyle\vartheta(A)\in\operatorname{im}(\delta),\quad\forall
A\in\mathbb{A}_{2},$

which we refer to as the DM condition. (This condition is not present for 3rd
order ODE.)

##  3\. Lowest weight vectors for irreducible C-class modules

The $\mathfrak{g}_{0}$-module structure for irreducible C-class modules
$\mathbb{U}\subset\mathbb{E}$ was stated in Table 2. While this abstract
structural information proved useful in our previous study of symmetry gaps
[13], more precise information is needed in our current study. Namely, viewing
$\mathbb{U}$ as harmonic 2-cochains via the $G_{0}$-equivariant identification
(2.18), we may ask for concrete realizations of lowest weight vectors
$\Phi_{\mathbb{U}}\in\mathbb{U}$ (from which a full basis of $\mathbb{U}$ may
be obtained by applying raising operators). These realizations are not found
in the existing literature, and our main goal in this section is to provide
them. This information will provide the starting point in subsequent sections
for our classification of submaximally symmetric structures.

Given the notation introduced in §2.2, and letting $E^{i,a}$ denote the dual
basis elements to $E_{i,a}$, we have:

######  Theorem 3.1.

Fix $n,m\geq 2$ and an irreducible C-class module
$\mathbb{U}\subset\mathbb{E}$, viewed as a $G_{0}$-submodule of
$\ker\square\subset C^{2}(\mathfrak{g}_{-},\mathfrak{g})$ via (2.18). Then the
unique lowest weight vector $\Phi_{\mathbb{U}}\in\mathbb{U}$, up to a scaling,
is given in Table 4.

$\begin{array}[]{|ccl|}\hline\cr
n&\begin{array}[]{c}\mathbb{U}\end{array}&\lx@intercol\hfil\mbox{Lowest weight
vector}\,\Phi_{\mathbb{U}}\in\mathbb{U}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr 2&\mathbb{B}_{4}&\begin{array}[]{l}E^{2,1}\wedge
E^{1,1}\otimes\mathsf{X}-\frac{1}{2}E^{2,1}\wedge
E^{0,1}\otimes\mathsf{H}-\frac{1}{2}E^{1,1}\wedge E^{0,1}\otimes\mathsf{Y}\\\
\quad+\sum_{a=1}^{m}\left(E^{2,1}\wedge E^{0,a}-E^{1,1}\wedge
E^{1,a}+E^{0,1}\wedge E^{2,a}\right)\otimes e_{a}{}^{1}\end{array}\\\
\hline\cr\geq
3&\mathbb{A}_{2}^{\operatorname{tr}}&\begin{array}[]{l}\alpha\sum_{i=0}^{n}\left[\Phi^{2,i}+(\frac{n}{2}-i)\Phi^{1,i}-\frac{1}{2}i(n+1-i)\Phi^{0,i}\right]\\\
\quad+\beta\sum_{i=0}^{n}\left[(n+1-i)(\Phi^{i,0}-\Phi^{0,i})+\Phi^{i,1}-\Phi^{1,i}\right],\\\\[3.61371pt]
\begin{array}[]{rl}\mbox{where}&\Phi^{i,j}:=\sum_{a=1}^{m}E^{i,1}\wedge
E^{j,a}\otimes E_{i+j-1,a}\\\
\mbox{and}&\alpha=\frac{-6(n-1)(m+1)}{mn(n+1)+6}\beta\end{array}\end{array}\\\
\hline\cr\geq
2&\mathbb{A}_{2}^{\operatorname{tf}}&\begin{array}[]{l}\sum_{j=0}^{n}[(n+1-j)\Phi^{0,j}+\Phi^{1,j}],\\\\[3.61371pt]
\begin{array}[]{rl}\mbox{where}&\Phi^{i,j}:=E^{i,1}\wedge E^{j,1}\otimes
E_{i+j-1,m}\end{array}\end{array}\\\ \hline\cr\end{array}$ Table 4.
Classification of lowest weight vectors $\Phi_{\mathbb{U}}$ for irreducible
C-class modules $\mathbb{U}\subset\mathbb{E}$

Let us give a brief summary of the computations to follow. For
$\Phi_{\mathbb{U}}\in\mathbb{U}$ lying in the appropriate module given in
(2.23), we use the bi-grade and $\mathfrak{sl}(W)$-lowest weight data for
$\mathbb{U}$ from Table 2 to first write a general form for
$\Phi_{\mathbb{U}}$. (The reader should recall the bi-grades given in §2.2,
e.g. $E_{i,a}$ has bi-grade $(-i,-1)$, and so
$E_{i,a}\in\mathfrak{g}_{-i-1}\subset\mathfrak{g}^{-i-1}$.) We then further
constrain this form by imposing additional linear conditions coming from
harmonicity, strong regularity, and the DM condition (2.27). (For example,
since $\Phi_{\mathbb{U}}\in\bigwedge^{2}V^{*}\otimes V$ in the
$\mathbb{A}_{2}^{\operatorname{tr}},\mathbb{A}_{2}^{\operatorname{tf}}$ cases,
then $\partial\Phi_{\mathbb{U}}=0$ if and only if
$\mathsf{X}\cdot\Phi_{\mathbb{U}}=0$ by Lemma 2.2. Imposing
$\mathsf{X}$-annihilation will be a detailed calculation involving the
relations $\mathsf{X}\cdot E_{i,a}=E_{i+1,a}$ and $\mathsf{X}\cdot
E^{i,a}=-E^{i-1,a}$.) This calculation will be involved, but we remark that in
fact not all such conditions will need to be explicitly imposed:

######  Remark 3.2.

If $\Phi_{\mathbb{U}}$ can be constrained to a 1-dimensional subspace by
imposing some of the conditions above, then $\Phi_{\mathbb{U}}$ necessarily
satisfies all the remaining linear conditions (harmonicity, strong regularity,
and (2.27)). This follows from existence of the module
$\mathbb{U}\subset\mathbb{E}$ for ODE systems, which was established in [18,
11].

Let us now carry out the indicated computations and establish Theorem 3.1
above.

###  3.1. $\mathbb{B}_{4}$ case

Since $\mathbb{U}:=\mathbb{B}_{4}\subset\bigwedge^{2}V^{*}\otimes\mathfrak{q}$
has bi-grade $(2,2)$, then $\Phi_{\mathbb{U}}$ must be a linear combination
of:

(3.1) $\displaystyle\begin{split}&E^{2,a}\wedge E^{1,b}\otimes\mathsf{X},\quad
E^{1,a}\wedge E^{0,b}\otimes\mathsf{Y},\quad E^{2,a}\wedge
E^{0,b}\otimes\mathsf{H},\quad E^{1,a}\wedge E^{1,b}\otimes\mathsf{H},\\\
&E^{2,a}\wedge E^{0,b}\otimes e_{c}{}^{d},\quad E^{1,a}\wedge E^{1,b}\otimes
e_{c}{}^{d}\quad(1\leq a,b,c,d\leq m).\end{split}$

Since $\mathbb{U}$ has $\mathfrak{sl}(W)$-lowest weight
$\lambda=-2\epsilon_{1}$ (Table 2), then $\Phi_{\mathbb{U}}$ lies in the
subspace spanned by

(3.2) $\displaystyle\begin{split}&E^{2,1}\wedge E^{1,1}\otimes\mathsf{X},\quad
E^{2,1}\wedge E^{0,1}\otimes\mathsf{H},\quad E^{1,1}\wedge
E^{0,1}\otimes\mathsf{Y},\quad E^{2,1}\wedge E^{0,1}\otimes e_{1}{}^{1},\\\
&E^{2,1}\wedge E^{0,1}\otimes e_{a}{}^{a},\quad E^{2,1}\wedge E^{0,a}\otimes
e_{a}{}^{1},\quad E^{2,a}\wedge E^{0,1}\otimes e_{a}{}^{1},\quad E^{1,a}\wedge
E^{1,1}\otimes e_{a}{}^{1}.\end{split}$

For $\mathfrak{ann}(\Phi_{\mathbb{U}})$ from Table 3, requiring
$\mathfrak{ann}(\Phi_{\mathbb{U}})\cdot\Phi_{\mathbb{U}}=0$ further constrains
$\Phi_{\mathbb{U}}$ to lie in span of:

(3.3) $\displaystyle\begin{split}&E^{2,1}\wedge E^{1,1}\otimes\mathsf{X},\quad
E^{2,1}\wedge E^{0,1}\otimes\mathsf{H},\quad E^{1,1}\wedge
E^{0,1}\otimes\mathsf{Y},\\\ &\sum_{a=1}^{m}E^{2,1}\wedge E^{0,1}\otimes
e_{a}{}^{a},\quad\sum_{a=1}^{m}E^{2,1}\wedge E^{0,a}\otimes
e_{a}{}^{1},\quad\sum_{a=1}^{m}E^{2,a}\wedge E^{0,1}\otimes
e_{a}{}^{1},\quad\sum_{a=1}^{m}E^{1,a}\wedge E^{1,1}\otimes
e_{a}{}^{1}.\end{split}$

Let us briefly explain this. From Table 3, $\mathfrak{sl}_{m-1}$ embeds into
$\mathfrak{ann}(\Phi_{\mathbb{U}})$ via $A\mapsto\operatorname{diag}(0,A)$,
which acts trivially on the first 4 elements of (3.2). The remaining tensors
in (3.2) lie in a direct sum of 4 $\mathfrak{sl}_{m-1}$-reps equivalent to the
sum of 4 copies of $\mathfrak{gl}_{m-1}$. (Namely, consider the span of
$E^{2,1}\wedge E^{0,1}\otimes e_{a}{}^{b},\,E^{2,1}\wedge E^{0,b}\otimes
e_{a}{}^{1}$, etc.) Since
$\mathfrak{gl}_{m-1}\cong\mathbb{R}\oplus\mathfrak{sl}_{m-1}$, then the
aforementioned subspace contains a 4-dimensional subspace annihilated by
$\mathfrak{sl}_{m-1}$. This is clearly spanned by the elements in the second
line of (3.3) except taking the sum over $2\leq a\leq m$. Finally, forcing
annihilation with respect to $e_{f}{}^{1}$ for $f\geq 2$ yields (3.3).

Let $\Phi_{\mathbb{U}}$ be a general linear combination of all elements of
(3.3), with $\mu_{i}$ denoting the coefficient of the $i$-th term, i.e.
$\Phi_{\mathbb{U}}=\mu_{1}E^{2,1}\wedge
E^{1,1}\otimes\mathsf{X}+\mu_{2}E^{2,1}\wedge
E^{0,1}\otimes\mathsf{H}+\ldots+\mu_{7}\sum_{a=1}^{m}E^{1,a}\wedge
E^{1,1}\otimes e_{a}{}^{1}$. We conclude our computation by imposing
$\partial$-closedness for $\Phi_{\mathbb{U}}$ using Lemma 2.2:

  * •

$\mathsf{X}$-annihilation: This yields $\mu_{2}=\mu_{3}=-\frac{\mu_{1}}{2}$,
$\mu_{4}=0$ and $\mu_{7}=\mu_{5}=-\mu_{6}$.

  * •

$\partial_{V}$-closedness:
$0=\partial_{V}\Phi_{\mathbb{U}}(E_{1,2},E_{2,1},E_{1,1})=(\mu_{5}-\mu_{1})E_{2,2}$,
and hence $\mu_{1}=\mu_{5}$.

This uniquely pins down $\Phi_{\mathbb{U}}$ (as stated in Table 4), up to a
nonzero scaling. From Remark 3.2, we in particular have that
$\Phi_{\mathbb{U}}$ is normal and strongly regular. (The condition (2.27) does
not apply for 3rd order ODE systems.)

###  3.2. $\mathbb{A}_{2}^{\operatorname{tr}}$ case

This case proceeds similarly, but is more involved than the $\mathbb{B}_{4}$
case. In particular, more conditions are required to pin down the lowest
weight vector (up to scale).

Let $n\geq 3$. Since
$\mathbb{U}:=\mathbb{A}_{2}^{\operatorname{tr}}\subset\bigwedge^{2}V^{*}\otimes
V$ has bi-grade $(1,1)$ and $\mathfrak{sl}(W)$-lowest weight
$\lambda=-\epsilon_{1}$, then $\Phi_{\mathbb{U}}$ must be a linear combination
of

(3.4) $\displaystyle E^{i,1}\wedge E^{j,a}\otimes E_{i+j-1,a}\quad(0\leq
i,j\leq n,\;1\leq i+j\leq n+1,\;1\leq a\leq m).$

Moreover, $\mathfrak{ann}(\Phi_{\mathbb{U}})$ from Table 3 annihilates
$\Phi_{\mathbb{U}}$, so $\Phi_{\mathbb{U}}$ is in fact constrained to be a
linear combination of

(3.5) $\displaystyle\Phi^{i,j}:=\sum_{a=1}^{m}E^{i,1}\wedge E^{j,a}\otimes
E_{i+j-1,a}.$

Recalling our convention in §2.2 that $E_{k}=0$ for $k<0$ or $k>n$, we have:

######  Proposition 3.3.

Fix $n\geq 3$ and $m\geq 2$. Let
$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tr}}$ and define
$\Phi_{\mathbb{U}}=\sum_{i,j=0}^{n}c_{i,j}\Phi^{i,j}$ for $\Phi^{i,j}$ as in
(3.5), where we may assume that $c_{0,0}=0=c_{i,j}$ for $i+j>n+1$. Since
$\Phi_{\mathbb{U}}$ is $\partial$-closed and satisfies the strong regularity
and DM conditions, then we have

(3.6) $\displaystyle\begin{cases}c_{i+1,j}+c_{i,j+1}=c_{i,j};&\mbox{(XA):
annihilation by $\mathsf{X}$;}\\\
c_{i,j}{}=0,\quad\mbox{for}\quad\min(i,j)\geq 3;&\mbox{(SR) : strong
regularity;}\\\ c_{n-1,2}=0,\quad\mbox{for}\quad n\geq 4.&\mbox{(DM): DM
conditions beyond (SR).}\end{cases}$

###### Proof.

By Lemma 2.2, $\partial$-closedness of $\Phi_{\mathbb{U}}$ is equivalent to
its $\mathsf{X}$-annihilation, so using $\mathsf{X}\cdot E_{i,a}=E_{i+1,a}$
and $\mathsf{X}\cdot E^{i,a}=-E^{i-1,a}$, Leibniz rule, and re-indexing the
summation, we straightforwardly obtain:

(3.7) $\displaystyle 0$
$\displaystyle=\mathsf{X}\cdot\Phi_{\mathbb{U}}=\sum_{i=0}^{n}\sum_{j=0}^{n}(c_{i,j}-c_{i+1,j}-c_{i,j+1})\Phi^{i,j}.$

This proves the first relations.

Next, recall that $E_{k,a}\in\mathfrak{g}^{-k-1}$. Strong regularity (2.14) of
$\Phi^{i,j}$ forces $E_{i+j-1,a}\in\mathfrak{g}^{\min(-i-1,-j-1)-1}$, i.e.

(3.8) $\displaystyle-i-j\geq\min(-i-1,-j-1)-1\geq-\max(i,j)-2\iff
i+j\leq\max(i,j)+2,$

or equivalently $\min(i,j)\leq 2$. All other terms are not present in the
summation.

Finally, for the last relations we force (2.27) for $A=\Phi_{\mathbb{U}}$,
i.e. $\vartheta(\Phi_{\mathbb{U}})\in\operatorname{im}(\delta)$. Recall the
maps $\delta$ and $\vartheta$ given in (2.25) and (2.26), and
$F=\operatorname{span}\\{E_{0},\ldots,E_{n-1}\\}\otimes W\subset V$. Modulo
$F$,

  * •

$\displaystyle\vartheta(\Phi_{\mathbb{U}})=\sum_{i,j=0}^{n}c_{i,j}\vartheta(\Phi^{i,j})\equiv\sum_{i,j=0}^{n}c_{i,j}\Phi^{i,j}|_{\bigwedge^{2}F}\equiv\sum_{i=2}^{n-1}c_{i,n+1-i}\Phi^{i,n+1-i}\stackrel{{\scriptstyle\mbox{\tiny(SR)}}}{{\equiv}}c_{2,n-1}\Phi^{2,n-1}+c_{n-1,2}\Phi^{n-1,2}$.

  * •

$\displaystyle\delta(E^{i,a}\otimes\mathsf{X})\equiv\sum_{b=1}^{m}E^{i,a}\wedge
E^{n-1,b}\otimes E_{n,b}$, i.e. bi-grade $(i-1,1)$ tensors for $0\leq i\leq
n-1$.

Since $\vartheta(\Phi_{\mathbb{U}})$ only consists of bi-grade $(1,1)$
tensors, it suffices to examine the $(1,1)$ subspace of
$\operatorname{im}(\delta)$. From above, this always contains $\Phi^{2,n-1}$
(modulo $F$), but does not contain $\Phi^{n-1,2}$ when $n\geq 4$. Hence,
beyond (SR), DM condition implies
$\vartheta(\Phi_{\mathbb{U}})\in\operatorname{im}(\delta)$, which forces
$c_{n-1,2}=0$ for $n\geq 4$. ∎

We now solve (3.6):

######  Proposition 3.4.

Fix $n\geq 3$. Then $(c_{i,j})_{0\leq i,j\leq n}$ from Proposition 3.3 is of
the following form:

(3.9) $\displaystyle\begin{split}&c_{i,0}=\begin{cases}(n-i+1)\beta,&3\leq
i\leq n;\\\ \alpha+(n-1)\beta,&i=2;\\\
\frac{n\alpha}{2}+(n-1)\beta,&i=1;\end{cases}\qquad
c_{0,i}=\begin{cases}-c_{1,0},&i=1;\\\ (i-n-1)(\beta+\frac{i\alpha}{2}),&2\leq
i\leq n\end{cases}\\\
&c_{1,i}=\left(\frac{n}{2}-i\right)\alpha+(\delta_{i}{}^{1}-1)\beta,\quad
1\leq i\leq n;\qquad c_{i,1}=\beta+\delta_{i}{}^{2}\alpha,\quad 2\leq i\leq
n;\\\ &c_{2,i}=(1-\delta_{i}{}^{n})\alpha,\quad 2\leq i\leq n;\hskip
85.35826ptc_{i,2}=0,\quad 3\leq i\leq n.\end{split}$

where $\alpha:=c_{2,n-1}$ and $\beta:=c_{n,1}$, and all other coefficients are
trivial.

###### Proof.

Since (DM) is only present for $n\geq 4$, we split our proof into two cases:

  * •

$n=3$: The system (3.6) becomes:

(3.13)
$\displaystyle\begin{array}[]{llll}c_{1,0}+c_{0,1}=0,&c_{1,1}+c_{0,2}=c_{0,1},&c_{2,0}+c_{1,1}=c_{1,0},\\\
c_{1,2}+c_{0,3}=c_{0,2},&c_{2,1}+c_{1,2}=c_{1,1},&c_{3,0}+c_{2,1}=c_{2,0},\\\
c_{1,3}=c_{0,3},&c_{2,2}+c_{1,3}=c_{1,2},&c_{3,1}+c_{2,2}=c_{2,1},&c_{3,1}=c_{3,0}.\end{array}$

Solving this in terms of $\alpha=c_{2,2}$ and $\beta=c_{3,1}$ gives (3.9).

  * •

$n\geq 4$:

Step 1: Start with the assumed conditions $c_{0,0}=0=c_{i,j}$ for $i+j>n+1$,
the (SR) relations, as well as the (DM) relation $c_{n-1,2}=0$. Using (XA),
determine the entries above $c_{n-1,2}=0$ and left of $c_{2,n-1}=:\alpha$
(until the $(2,2)$-position), as shown below.

(3.34)
$\displaystyle\tiny\left(\begin{array}[]{ccc|cccccccc}0&*&*&*&*&\cdots&*&*&*&*\\\
&*&*&*&*&\cdots&*&*&*&*\\\
&*&*&*&*&\cdots&*&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}&0\\\
\hline\cr*&*&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\cdots}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&0\\\
\vdots&\vdots&\vdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\cdots}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&0&0\\\
\vdots&\vdots&\vdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&\reflectbox{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
$\ddots$}}&0&0&0&0\\\
&*&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&0&0&0&0\\\
&*&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&0&0&0&0&0\\\
&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&0&0&0&0&0&0\\\
&*&0&0&0&0&0&0&0&0\end{array}\right)\quad\leadsto\quad\left(\begin{array}[]{ccc|cccccccc}0&*&*&*&*&\cdots&*&*&*&*\\\
&*&*&*&*&\cdots&*&*&*&*\\\
&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}&\cdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha}&0\\\
\hline\cr*&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&0&\cdots&0&0&0&0\\\
\vdots&\vdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vdots}&0&0&\cdots&0&0&0&0\\\
\vdots&\vdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vdots}&0&0&\reflectbox{$\ddots$}&0&0&0&0\\\
&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&0&0&0&0&0&0\\\
&*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&0&0&0&0&0&0\\\
&*&0&0&0&0&0&0&0&0\\\ &*&0&0&0&0&0&0&0&0\end{array}\right)$

Step 2: Using (XA), we have $\beta:=c_{n,1}=c_{n,0}$. Use (XA) to determine
the entries above $c_{n,1}$ and left of $\gamma:=c_{1,n}$ (until the
$(1,1)$-position), as shown below.

(3.45)
$\displaystyle\tiny\left(\begin{array}[]{ccc|cccccccc}0&*&*&*&*&\cdots&*&*&*&*\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(n-1)\alpha+\beta+\gamma}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(n-2)\alpha+\gamma}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(n-3)\alpha+\gamma}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(n-4)\alpha+\gamma}&\cdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3\alpha+\gamma}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2\alpha+\gamma}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha+\gamma}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma}\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\alpha+\beta}&\alpha&\alpha&\alpha&\cdots&\alpha&\alpha&\alpha&0\\\
\hline\cr*&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta}&0&0&0&\cdots&0&0&0&0\\\
\vdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vdots}&\vdots&0&0&\cdots&0&0&0&0\\\
\vdots&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vdots}&\vdots&0&0&\reflectbox{$\ddots$}&0&0&0&0\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta}&0&0&0&0&0&0&0&0\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta}&0&0&0&0&0&0&0&0\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta}&0&0&0&0&0&0&0&0\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta}&0&0&0&0&0&0&0&0\end{array}\right).$

More precisely, we have

(3.46) $\displaystyle c_{1,i}=(n-i)\alpha+\delta_{i}^{1}\beta+\gamma,\quad
1\leq i\leq n.$

Step 3: Using (XA), determine all entries above $c_{n,0}=\beta$. This yields:

(3.47) $\displaystyle c_{i,0}=\begin{cases}(n-i+1)\beta,&3\leq i\leq n;\\\
\alpha+(n-1)\beta,&i=2;\\\ n(\alpha+\beta)+\gamma,&i=1.\end{cases}$

Step 4: Impose
$c_{0,1}\stackrel{{\scriptstyle\mbox{\tiny(XA)}}}{{=}}-c_{1,0}=-n(\alpha+\beta)-\gamma$.
For $2\leq i\leq n$, we have the telescoping sum:

(3.48) $\displaystyle c_{0,i}-c_{0,1}$
$\displaystyle=\sum_{k=2}^{i}(c_{0,k}-c_{0,k-1})\stackrel{{\scriptstyle\mbox{\tiny(XA)}}}{{=}}-\sum_{k=2}^{i}c_{1,k-1}\stackrel{{\scriptstyle\eqref{E:c1i}}}{{=}}-\sum_{k=2}^{i}[(n-k+1)\alpha+\delta_{k-1}^{1}\beta+\gamma]$
(3.49)
$\displaystyle\begin{split}c_{0,i}&=c_{0,1}-\beta-(i-1)\gamma-\alpha[(n-1)+...+(n-i+1)]\\\
&=-(n+1)\beta-i\gamma-\alpha\left[\binom{n+1}{2}-\binom{n-i+1}{2}\right]\end{split}$

Step 5: Impose
$c_{0,n}\stackrel{{\scriptstyle\mbox{\tiny(XA)}}}{{=}}c_{1,n}=\gamma$. Solving
this yields $\gamma=-\beta-\frac{n\alpha}{2}$. Substituting this into (3.46),
(3.47) and (3.49) then gives the stated result.

∎

We conclude our computation by imposing the coclosedness condition, i.e.
$\partial^{\ast}\Phi_{\mathbb{U}}=0$.

######  Proposition 3.5.

Let $n\geq 3$ and $m\geq 2$. Take $\Phi_{\mathbb{U}}$ from Proposition 3.3
with coefficients (3.9). Then:

(3.50) $\displaystyle\alpha=\frac{-6(n-1)(m+1)}{mn(n+1)+6}\beta.$

###### Proof.

From Lemma B.2, we have:

(3.51) $\displaystyle
0=\sum_{k=0}^{n-1}\frac{(n-k)(k+1)}{n(n-1)}(c_{k,2}-mc_{2,k})+\sum_{k=0}^{n}\frac{2k-n}{n}(mc_{1,k}-c_{k,1})+\sum_{k=1}^{n}(mc_{0,k}-c_{k,0}).$

We now substitute (3.9) into (3.51) and simplify. The computations are
straightforward but tedious, and for the respective summations above, this
leads to:

(3.52) $\displaystyle 0$
$\displaystyle=-\frac{(n+2)\Omega}{6n(n-1)}-\frac{(n+2)\Omega}{6n}-\frac{(n+2)\Omega}{12}=-\frac{(n+2)(n+1)\Omega}{12(n-1)},$

where $\Omega:=(mn(n+1)+6)\alpha+6(n-1)(m+1)\beta$. This implies $\Omega=0$,
and hence the result. ∎

Combining (3.50), (3.9), and Proposition 3.3 uniquely determines
$\Phi_{\mathbb{U}}$ (given in Table 4 after some simplification /
reorganization), up to a nonzero scaling. Note that (3.51) (derived in
Appendix B) was only a small part of the coclosedness condition, but using
Remark 3.2, we deduce that indeed $\partial^{*}\Phi_{\mathbb{U}}=0$.

###  3.3. $\mathbb{A}_{2}^{\operatorname{tf}}$ case

The trace-free case proceeds analogously to the trace case. Since
$\mathbb{U}:=\mathbb{A}_{2}^{\operatorname{tf}}\subset\bigwedge^{2}V^{*}\otimes
V$ has bi-grade $(1,1)$ and $\mathfrak{sl}(W)$-lowest weight
$\lambda=\epsilon_{m}-2\epsilon_{1}$, then $\Phi_{\mathbb{U}}$ is a linear
combination of

(3.53) $\displaystyle\Phi^{i,j}:=E^{i,1}\wedge E^{j,1}\otimes
E_{i+j-1,m}\quad(0\leq i,j\leq n,\quad 1\leq i+j\leq n+1).$

Note that $\Phi^{i,j}=-\Phi^{j,i}$ is annihilated by all elements of
$\mathfrak{ann}(\Phi_{\mathbb{U}})$ given in Table 3.

######  Proposition 3.6.

Fix $n,m\geq 2$. Let $\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tf}}$ and
define $\Phi_{\mathbb{U}}=\sum_{i,j=0}^{n}c_{i,j}\Phi^{i,j}$ for $\Phi^{i,j}$
as in (3.53), where we may assume that $c_{i,j}=-c_{j,i}$, and $c_{i,j}=0$ for
$i+j>n+1$. Since $\Phi_{\mathbb{U}}$ is $\partial$-closed and satisfies the
strong regularity and DM conditions, then we have

(3.54) $\displaystyle\begin{cases}c_{i+1,j}+c_{i,j+1}=c_{i,j};&\mbox{(XA):
annihilation by $\mathsf{X}$;}\\\
c_{i,j}{}=0,\quad\mbox{for}\quad\min(i,j)\geq 3;&\mbox{(SR): strong
regularity;}\\\ c_{n-1,2}=c_{2,n-1}=0,\quad\mbox{for}\quad\framebox{$n\geq
3$}.&\mbox{(DM'): DM conditions beyond (SR).}\end{cases}$

###### Proof.

The proof is very similar to Proposition 3.3, as we now explain. Recall that
$\mathsf{X}\cdot E_{i,a}=E_{i+1,a}$ and $\mathsf{X}\cdot E^{i,a}=-E^{i-1,a}$,
i.e. the $\mathsf{X}$-action on these basis elements is independent of the
second index. Consequently, comparing (3.53) and (3.5), it is immediate that
$\mathsf{X}\cdot\Phi_{\mathbb{U}}=0$ yields the same conditions (XA). Strong
regularity similarly does not involve the second index, and so we obtain the
same conditions (SR).

Finally, let us focus on (2.27). As in the proof of Proposition 3.3,

  * •

$\displaystyle\vartheta(\Phi_{\mathbb{U}})\equiv\sum_{i=2}^{n-1}c_{i,n+1-i}\Phi^{i,n+1-i}\mod
F$, which consists of bi-grade $(1,1)$ tensors;

  * •

the bi-grade $(1,1)$ tensors in $\operatorname{im}(\delta)$ are spanned by
$\displaystyle\sum_{b=1}^{m}E^{2,a}\wedge E^{n-1,b}\otimes E_{n,b}\mod F$.

Since $m\geq 2$, then
$\vartheta(\Phi_{\mathbb{U}})\in\operatorname{im}(\delta)$ forces
$\vartheta(\Phi_{\mathbb{U}})\equiv 0$. This is automatic for $n=2$, while for
$n\geq 3$, we have $c_{i,n+1-i}=0$ for $2\leq i\leq n-1$. Beyond (SR), we have
merely $c_{2,n-1}=c_{n-1,2}=0$. ∎

######  Proposition 3.7.

Fix $n\geq 2$. Then $(c_{i,j})_{0\leq i,j\leq n}$ from Proposition 3.6 is of
the following form:

(3.55) $\displaystyle c_{i,0}=-c_{0,i}=\begin{cases}(n-i+1)\beta,&2\leq i\leq
n\\\ (n-1)\beta,&i=1\end{cases}\qquad\mbox{and}\qquad
c_{i,1}=-c_{1,i}=\beta,\quad 2\leq i\leq n$

and all other coefficients are trivial.

###### Proof.

We split our proof into two cases:

  * •

$n=2$: the system (3.54) reduces to

(3.56) $\displaystyle c_{1,0}+c_{0,1}=0,\quad c_{2,0}=c_{1,0},\quad
c_{0,2}=c_{0,1},\quad c_{1,2}=c_{0,2},\quad c_{2,1}=c_{2,0},$

and solving the system in terms of $c_{2,1}$ proves the claim.

  * •

$n\geq 3$: The conditions on $c_{i,j}$ in Proposition 3.6 can be viewed as
(3.6) with additionally $c_{i,j}=-c_{j,i}$ (and $c_{2,n-1}=0$ when $n=3$).
Consequently, the solution to (3.54) can be obtained from the solution (3.9)
to (3.6) by merely imposing $\alpha:=c_{2,n-1}=0$.

∎

Combining (3.55) and Proposition 3.6 uniquely determines $\Phi_{\mathbb{U}}$
(given in Table 4), up to a nonzero scaling. As before, using Remark 3.2, we
deduce that $\partial^{*}\Phi_{\mathbb{U}}=0$. This completes our proof of
Theorem 3.1.

##  4\. Homogeneous structures and Cartan-theoretic descriptions

Our method for proving Theorems 1.1 and 1.2 will rely on the fact that Cartan
geometries (see §2.3) associated to submaximally symmetric vector ODEs are
locally homogeneous. In this section, we summarize all relevant symmetry-based
facts about such geometries and their corresponding algebraic models of ODE
type. We will use $G,P,G_{0}$ and $\mathfrak{g}$ from §2.2, and the filtration
and grading on $\mathfrak{g}$ defined there.

###  4.1. Symmetry gaps for ODE

An infinitesimal symmetry of a given Cartan geometry
$(\mathcal{G}\to\mathcal{E},\omega)$ of type $(G,P)$ is a $P$-invariant vector
field $\xi\in\mathfrak{X}(\mathcal{G}){}^{P}$ on $\mathcal{G}$ that preserves
$\omega$ under Lie differentiation, i.e. $\mathcal{L}_{\xi}\omega=0$. The
collection of all such vector fields forms a Lie algebra, which we denote by

(4.1)
$\displaystyle\mathfrak{inf}(\mathcal{G},\omega):=\\{\xi\in\mathfrak{X}(\mathcal{G}){}^{P}\,:\,\mathcal{L}_{\xi}\omega=0\\}\subset\mathfrak{X}(\mathcal{G}).$

The submaximal symmetry dimension is

(4.2)
$\displaystyle\begin{split}&\mathfrak{S}:=\max\left\\{\dim\mathfrak{inf}(\mathcal{G},\omega)\,:\,(\mathcal{G}\to\mathcal{E},\omega)\quad\mbox{strongly
regular, normal of type}\quad(G,P)\right.\\\ &\left.\hskip
142.26378pt\mbox{associated to a vector ODE $\mathcal{E}$
\eqref{ODE},}\quad\mbox{with}\quad\kappa_{H}\not\equiv 0\right\\}.\end{split}$

Recall that $\mathbb{E}$ decomposes into $G_{0}$-irreducible submodules
$\mathbb{U}\subset\mathbb{E}$. Analogous to $\mathfrak{S}$ above, we define:

(4.3)
$\displaystyle\mathfrak{S}_{\mathbb{U}}:=\max\left\\{\dim\mathfrak{inf}(\mathcal{G},\omega)\,:\,(\mathcal{G}\to\mathcal{E},\omega)\quad\mbox{strongly
regular, normal of type}\quad(G,P)\right.$ $\displaystyle\left.\hskip
142.26378pt\mbox{associated to a vector ODE $\mathcal{E}$
\eqref{ODE},}\quad\mbox{with}\quad
0\not\equiv\operatorname{im}(\kappa_{H})\subset\mathbb{U}\right\\}.$

In order to define suitable algebraic upper bounds, we will need the following
notion from [15]:

######  Definition 4.1.

Given a subspace $\mathfrak{a}_{0}\subset\mathfrak{g}_{0}$, the graded
subalgebra
$\mathfrak{a}=\operatorname{pr}(\mathfrak{g}_{-},\mathfrak{a}_{0}):=\mathfrak{a}_{-}\oplus\mathfrak{a}_{0}\oplus\mathfrak{a}_{1}\subset\mathfrak{g}$,
where
$\mathfrak{a}_{-}:=\mathfrak{g}_{-}=\mathfrak{g}_{-n-1}\oplus\ldots\oplus\mathfrak{g}_{-1}$
and
$\mathfrak{a}_{1}:=\\{x\in\mathfrak{g}_{1}\,:\,[x,\mathfrak{g}_{-1}]\subset\mathfrak{a}_{0}\\}$
is called the Tanaka prolongation algebra. For $\phi$ in some
$\mathfrak{g}_{0}$-module, we define
$\mathfrak{a}^{\phi}:=\operatorname{pr}(\mathfrak{g}_{-},\mathfrak{ann}(\phi))$,
where $\mathfrak{ann}(\phi)\subset\mathfrak{g}_{0}$ is the annihilator of
$\phi$.

Now, we define

(4.4)
$\displaystyle\mathfrak{U}:=\max\left\\{\dim\mathfrak{a}^{\phi}\,:\,0\neq\phi\in\mathbb{E}\right\\}\quad\mbox{and}\quad\mathfrak{U}_{\mathbb{U}}:=\max\left\\{\dim\mathfrak{a}^{\phi}\,:\,0\neq\phi\in\mathbb{U}\right\\}.$

By [13, Thm 2.11], we conclude that

(4.5)
$\displaystyle\mathfrak{S}\leq\mathfrak{U}<\dim\mathfrak{g}\quad\mbox{and}\quad\mathfrak{S}_{\mathbb{U}}\leq\mathfrak{U}_{\mathbb{U}}\quad\mbox{for
all $G_{0}$- irreducible modules}\quad\mathbb{U}\subset\mathbb{E}.$

Note that
$\mathfrak{U}=\max_{\mathbb{U}\subset\mathbb{E}}\mathfrak{U}_{\mathbb{U}}$. In
fact, by [13, Thm 1.2], in all of the vector cases we have equality:

(4.6)
$\displaystyle\mathfrak{S}=\mathfrak{U}\qquad\mbox{and}\qquad\mathfrak{S}_{\mathbb{U}}=\mathfrak{U}_{\mathbb{U}}.$

Examples of some vector ODEs realizing these can be found in [13, Table 8 &
10].

###  4.2. Local homogeneity and algebraic models of ODE type

Recall that a Cartan geometry $(\mathcal{G}\to\mathcal{E},\omega)$ of type
$(G,P)$ is said to be locally homogeneous if there exists a (left) action by a
local Lie group $F$ on $\mathcal{G}$ by principal bundle morphisms preserving
$\omega$ that projects onto a transitive action down on $\mathcal{E}$. We then
have [13, Lemma A.1]:

######  Lemma 4.2.

Fix a $G_{0}$-irrep $\mathbb{U}\subset\mathbb{E}$. Then any regular, normal
Cartan geometry $(\mathcal{G}\to\mathcal{E},\omega)$ of type $(G,P)$ with
$0\not\equiv\operatorname{im}(\kappa_{H})\subset\mathbb{U}$ and
$\dim(\mathfrak{inf}(\mathcal{G},\omega))=\mathfrak{U}_{\mathbb{U}}$ is
locally homogeneous about any point $u\in\mathcal{G}$ with $\kappa_{H}(u)\neq
0$.

By [13, §A.1], such a homogeneous Cartan geometry can be encoded Cartan-
theoretically by:

######  Definition 4.3.

An algebraic model $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type is a
Lie algebra $(\mathfrak{f},[\cdot,\cdot]_{\mathfrak{f}})$ such that:

  * (i)

$\mathfrak{f}\subset\mathfrak{g}$ is a filtered subspace whose associated
graded $\mathfrak{s}:=\operatorname{gr}(\mathfrak{f})\subset\mathfrak{g}$ has
$\mathfrak{s}_{-}=\mathfrak{g}_{-}$;

  * (ii)

$\mathfrak{f}^{0}$ inserts trivially into
$\kappa(x,y):=[x,y]-[x,y]_{\mathfrak{f}}$, i.e. $\kappa(z,\cdot)=0$ for all
$z\in\mathfrak{f}^{0}$;

  * (iii)

$\kappa$ is normal and strongly regular : $\partial^{\ast}\kappa=0$ and
$\kappa(\mathfrak{g}^{i},\mathfrak{g}^{j})\subset\mathfrak{g}^{i+j+1}\cap\mathfrak{g}^{\min(i,j)-1},\,\forall
i,j$.

Let $\mathcal{N}$ denote the set of all algebraic models
$(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type for fixed $(G,P)$. Then
$\mathcal{N}$ admits a $P$-action and is partially ordered:

  1. (1)

$P$-action: for $p\in P$ and $\mathfrak{f}\in\mathcal{N}$, we have
$p\cdot\mathfrak{f}:=\operatorname{Ad}_{p}(\mathfrak{f})$. We will regard all
algebraic models $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type in the
same $P$-orbit to be equivalent.

  2. (2)

partial order relation $\leq$: for
$\mathfrak{f},\widetilde{\mathfrak{f}}\in\mathcal{N}$ regard
$\mathfrak{f}\leq\widetilde{\mathfrak{f}}$ if there exists a map
$\mathfrak{f}\hookrightarrow\widetilde{\mathfrak{f}}$ of Lie algebras. We will
focus on maximal elements in $(\mathcal{N},\leq)$.

Combining (4.5), Lemma 4.2, and Definition 4.3, we obtain the following key
existence result:

######  Theorem 4.4.

Fix an irreducible $G_{0}$-module $\mathbb{U}$ in the effective part
$\mathbb{E}$ for vector ODEs (1.2) of order $\geq 3$. Then there exists an
algebraic model $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type with
$0\not\equiv\operatorname{im}(\kappa_{H})\subset\mathbb{U}$ and
$\dim\mathfrak{f}=\mathfrak{U}_{\mathbb{U}}=\mathfrak{S}_{\mathbb{U}}$.

######  Remark 4.5.

Conversely, by [15, Lemma 4.1.4], for a given algebraic model
$(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type, there exists a locally
homogeneous strongly regular, normal Cartan geometry
$(\mathcal{G}\to\mathcal{E},\omega)$ of type $(G,P)$ with
$\mathfrak{inf}(\mathcal{G},\omega)$ containing a subalgebra isomorphic to
$\mathfrak{f}$.

We caution that such a geometry may not arise from an ODE (1.2). (For
instance, the Doubrov–Medvedev condition must additionally hold.)
Consequently, our strategy involves:

  * •

classifying (up to the $P$-action) the corresponding algebraic models
$(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type, and then

  * •

providing vector ODEs of C-class realizing these algebraic models.

We will use the following results from [13, §A.1] in carrying out the
classifications.

######  Definition 4.6.

A filtered linear space $\mathfrak{f}\subset\mathfrak{g}$ can be described as
the graph of some linear map on $\mathfrak{s}$ into $\mathfrak{g}$ as follows.
Let $\mathfrak{s}^{\perp}\subset\mathfrak{g}$ be a graded subalgebra such that
$\mathfrak{g}=\mathfrak{s}\oplus\mathfrak{s}^{\perp}$. Then
$\mathfrak{f}:=\bigoplus_{i}\operatorname{span}\left\\{x+\mathfrak{D}(x):x\in\mathfrak{s}_{i}\right\\}$,
for some unique linear (deformation) map
$\mathfrak{D}:\mathfrak{s}\to\mathfrak{s}^{\perp}$ such that
$\mathfrak{D}(x)\in\mathfrak{s}^{\perp}\cap\mathfrak{g}^{i+1}$ for
$x\in\mathfrak{s}_{i}$.

######  Lemma 4.7.

Let $T\in\mathfrak{f}^{0}$ and suppose that the complementary graded subspaces
$\mathfrak{s},\mathfrak{s}^{\perp}\subset\mathfrak{g}$ are
$\operatorname{ad}_{T}$-invariant, then the map
$\mathfrak{D}:\mathfrak{s}\to\mathfrak{s}^{\perp}$ is
$\operatorname{ad}_{T}$-invariant, i.e.
$T\cdot\mathfrak{D}=0\iff\,\operatorname{ad}_{T}\circ\mathfrak{D}=\mathfrak{D}\circ\operatorname{ad}_{T}$.

Recall from §2.3 that $\kappa_{H}:=\kappa\mod\operatorname{im}\partial^{*}$,
where $\partial^{*}$ is the codifferential. We then have:

######  Proposition 4.8.

Let $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ be an algebraic model of ODE
type. Then

  * (i)

$(\mathfrak{f},[\cdot,\cdot]_{\mathfrak{f}})$ is a filtered Lie algebra.

  * (ii)

$\mathfrak{f}^{0}$ annihilates $\kappa$, i.e.
$\mathfrak{f}^{0}\cdot\kappa=0\iff[z,\kappa(x,y)]_{\mathfrak{f}}=\kappa([z,x]_{\mathfrak{f}},y)+\kappa(x,[z,y]_{\mathfrak{f}}),\,\forall
x,y\in\mathfrak{f},\,\forall z\in\mathfrak{f}^{0}$.

  * (iii)

$\mathfrak{s}\subset\mathfrak{a}^{\kappa_{H}}$.

Following [23, §2.2], we shall refer to $\mathfrak{f}$ as a (constrained)
filtered sub-deformation of $\mathfrak{s}$.

###  4.3. Characterizing maximality of the Tanaka prolongation

Fix a $G_{0}$-irrep $\mathbb{U}\subset\mathbb{E}$, and recall
$\mathfrak{U}_{\mathbb{U}}$ defined in (4.4). For vector ODEs (1.2),
$\mathfrak{U}_{\mathbb{U}}$ were computed in [13, §3.4] using the fact that
$\mathfrak{U}_{\mathbb{U}}=\dim\mathfrak{a}^{\Phi_{\mathbb{U}}}$, where
$\Phi_{\mathbb{U}}\in\mathbb{U}$ is an extremal (lowest or highest) weight
vector. For the purpose of our goal in §5, we next prove that
$\mathfrak{U}_{\mathbb{U}}$ is achieved precisely in this way:

######  Lemma 4.9.

Let $\mathbb{U}\subset\mathbb{E}$ be a $G_{0}$-irrep and
$\Phi_{\mathbb{U}}\in\mathbb{U}$ be a lowest weight vector. Then,
$\mathfrak{U}_{\mathbb{U}}=\dim\mathfrak{a}^{\Phi_{\mathbb{U}}}$. Moreover, if
$0\neq\phi\in\mathbb{U}$, then
$\dim\mathfrak{a}^{\phi}=\mathfrak{U}_{\mathbb{U}}$ iff $[\phi]$ is contained
in the $G_{0}$-orbit of $[\Phi_{\mathbb{U}}]\in\mathbb{P}(\mathbb{U})$.

###### Proof.

The proof used in [15, Prop 3.1.1] can be applied for our purposes here. (We
note that the initial hypothesis of $G$ complex semisimple Lie group and
$P\leq G$ a parabolic subgroup is not necessary. We use our $G_{0}$ here for
the $G_{0}$ appearing there.) Over $\mathbb{C}$, the same proof yields the
result. Over $\mathbb{R}$, the essential fact used in the proof is that
$\operatorname{SL}_{m}\subset G_{0}$ acts with a unique closed orbit
$\mathcal{O}$ (of minimal dimension) in $\mathbb{P}(\mathbb{U})$. We can
directly verify this for the modules $\mathbb{U}\subset\mathbb{E}$ in Table 2:

$\begin{array}[]{ccc}\mathbb{U}&\mathfrak{sl}(W)\mbox{-module
structure}&\mathcal{O}\subset\mathbb{P}(\mathbb{U})\\\
\hline\cr\mathbb{B}_{4}&S^{2}W^{*}&\\{[\eta^{2}]:[\eta]\in\mathbb{P}(W^{*})\\}\\\
\mathbb{A}_{2}^{\operatorname{tr}}&W^{*}&\mathbb{P}(W^{*})\\\
\mathbb{A}_{2}^{\operatorname{tf}}&(S^{2}W^{*}\otimes
W)_{0}&\\{[\eta^{2}\otimes
w]:[\eta]\in\mathbb{P}(W^{*}),\,[w]\in\mathbb{P}(W),\,\eta(w)=0\\}\\\
\hline\cr\end{array}$

∎

###  4.4. Prolongation-rigidity

In terms of the Tanaka prolongation algebra $\mathfrak{a}^{\phi}$ (Definition
4.1), we define:

######  Definition 4.10.

A $G_{0}$-module $\mathbb{U}\subset\mathbb{E}$ is said to be prolongation-
rigid (PR) if $\mathfrak{a}_{1}^{\phi}=0$ for all non-zero
$\phi\in\mathbb{U}$.

Let $\mathbb{U}\subset\mathbb{E}$ be an irreducible C-class module (see §2.5).
To study prolongation-rigidity, it suffices by Lemma 4.9 to consider the
lowest weight vector $\phi=\Phi_{\mathbb{U}}\in\mathbb{U}$. By [13, Lemma
3.3], we have $\mathfrak{a}^{\Phi_{\mathbb{U}}}_{1}=\mathbb{R}\mathsf{Y}$ if
and only if $\mathbb{U}$ has bi-grade that is a multiple of $(n,2)$. From
Table 2, the bi-grade of $\mathbb{U}$ is a multiple of $(1,1)$, so
$\mathbb{U}$ is not PR if and only if $n=2$. A summary is given in Table 5,
with $\mathfrak{a}^{\Phi_{\mathbb{U}}}$ in each case, and
$\mathfrak{ann}(\Phi_{\mathbb{U}})$ stated in Table 3.

$\begin{array}[]{|cccl|}\hline\cr n&\mathbb{U}&\mathbb{U}\mbox{
PR?}&\lx@intercol\hfil\mathfrak{a}^{\Phi_{\mathbb{U}}}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr
2&\mathbb{B}_{4}&\times&\mathfrak{g}_{-}\oplus\mathfrak{ann}(\Phi_{\mathbb{U}})\oplus\mathbb{R}\mathsf{Y}\\\
\geq
3&\mathbb{A}_{2}^{\operatorname{tr}}&\checkmark&\mathfrak{g}_{-}\oplus\mathfrak{ann}(\Phi_{\mathbb{U}})\\\
\hline\cr
2&\mathbb{A}_{2}^{\operatorname{tf}}&\times&\mathfrak{g}_{-}\oplus\mathfrak{ann}(\Phi_{\mathbb{U}})\oplus\mathbb{R}\mathsf{Y}\\\
\geq
3&\mathbb{A}_{2}^{\operatorname{tf}}&\checkmark&\mathfrak{g}_{-}\oplus\mathfrak{ann}(\Phi_{\mathbb{U}})\\\
\hline\cr\end{array}$ Table 5. Prolongation-rigidity for irreducible C-class
modules $\mathbb{U}\subset\mathbb{E}$

##  5\. Embeddings of filtered sub-deformations

By §4.2 above, all submaximally symmetric vector ODEs (1.2) can be encoded
using algebraic models of ODE type. Consequently, proving our main results
(Theorems 1.1 and 1.2) boils down to classifying these corresponding algebraic
models (Theorem 4.4). More precisely, in view of Lemma 4.9, for each
irreducible C-class module
$\mathbb{U}\subset\mathbb{E}_{C}\subsetneq\mathbb{E}$ (Definition 2.7), our
goal is to classify (up to the $P$-action) all algebraic models
$(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type with
$\kappa_{H}=\Phi_{\mathbb{U}}\in\mathbb{U}$, where $\Phi_{\mathbb{U}}$ is the
lowest weight vector from Table 4, and
$\dim\mathfrak{f}=\mathfrak{S}_{\mathbb{U}}$.

In this section, we classify all possible (filtered) linear embeddings
$\mathfrak{f}\subset\mathfrak{g}$ for such
$(\mathfrak{f};\mathfrak{g},\mathfrak{p})$. The possibilities for curvature
$\kappa$ of $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ are then classified in
§6. Recall the canonical filtration and the grading structure on
$\mathfrak{g}$ from §2.2. Having computed graded subalgebras
$\mathfrak{a}^{\Phi_{\mathbb{U}}}\subset\mathfrak{g}$ in Table 5, we next
classify, up to the $P$-action, possible filtered linear subspaces
$\mathfrak{f}\subset\mathfrak{g}$ for algebraic models
$(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ satisfying
$\operatorname{gr}(\mathfrak{f})=\mathfrak{a}^{\Phi_{\mathbb{U}}}$:

######  Proposition 5.1.

Fix an irreducible C-class module
$\mathbb{U}=\mathbb{B}_{4},\,\mathbb{A}_{2}^{\operatorname{tf}}$ or
$\mathbb{A}_{2}^{\operatorname{tr}}$ in $\mathbb{E}_{C}\subsetneq\mathbb{E}$,
viewed as a $G_{0}$-submodule of $\ker\square\subset
C^{2}(\mathfrak{g}_{-},\mathfrak{g})$ via (2.18), and consider an algebraic
model $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type with
$\kappa_{H}=\Phi_{\mathbb{U}}\in\mathbb{U}$, for $\Phi_{\mathbb{U}}$ from
Table 4, and $\dim\mathfrak{f}=\mathfrak{S}_{\mathbb{U}}$. Then using the
$P$-action, $\mathfrak{f}\mapsto\operatorname{Ad}_{p}\mathfrak{f}$, we may
normalize to

  * (a)

$\mathfrak{f}=\mathfrak{a}^{\Phi_{\mathbb{U}}}$ when
$\mathbb{U}=\mathbb{B}_{4}$ or $\mathbb{A}_{2}^{\operatorname{tf}};$

  * (b)

$\mathfrak{f}=\operatorname{span}\\{E_{n,a},\,\ldots,\,E_{2,a},\,E_{1,1}+(n-2)\zeta\mathsf{Z}_{1},\,E_{1,b},\,E_{0,1}+\zeta\mathsf{Y},\,E_{0,b},\,\mathsf{X}\,:\,\zeta\in\mathbb{R},\,1\leq
a\leq m,\,2\leq b\leq
m\\}\oplus\mathfrak{ann}(\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}})$ when
$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tr}}$.

###### Proof.

Since $\mathbb{U}$ has a bi-grade that is a multiple of $(1,1)$ (see Table 2),
then
$T:=\mathsf{Z}_{1}-\mathsf{Z}_{2}\in\mathfrak{ann}(\Phi_{\mathbb{U}})=\mathfrak{a}^{\Phi_{\mathbb{U}}}_{0}$.
We note that $\mathfrak{a}^{\Phi_{\mathbb{U}}}\subset\mathfrak{g}$ is a graded
subalgebra, and denote by $\widehat{T}$ the element in $\mathfrak{f}^{0}$ with
the leading part $T$, i.e. $\operatorname{gr}_{0}(\widehat{T})=T$. Since
$\mathfrak{g}_{1}=\mathbb{R}\mathsf{Y}$ and $\mathfrak{g}_{i}=0$ for all
$i\geq 2$ (see §2.2), then necessarily
$\widehat{T}=T+s\mathsf{Y}\in\mathfrak{f}^{0}$. We claim that without loss of
generality, i.e. using the $P$-action (for $P$ defined in §2.2), we may assume
that $T\in\mathfrak{f}^{0}$. This is immediate when $\mathbb{U}$ is not PR,
since $\mathsf{Y}\in\mathfrak{f}^{0}$ and therefore
$T=\widehat{T}-s\mathsf{Y}$ is a linear combination of $\widehat{T}$ and
$\mathsf{Y}$. Otherwise, when $\mathbb{U}$ is PR
($\mathsf{Y}\not\in\mathfrak{f}^{0}$), using the $P_{+}$-action and
$[\mathsf{Y},T]=-\mathsf{Y}$, we have:

(5.1) $\displaystyle\operatorname{Ad}_{\exp(t\mathsf{Y})}(\widehat{T})$
$\displaystyle=\exp(\operatorname{ad}_{t\mathsf{Y}})(\widehat{T})=\widehat{T}+[t\mathsf{Y},\widehat{T}]+\frac{1}{2!}[t\mathsf{Y},[t\mathsf{Y},\widehat{T}]]+\cdots=\mathsf{Z}_{1}-\mathsf{Z}_{2}+(s-t)\mathsf{Y},$

then choosing $t=s$ normalizes the right-hand side to $T$. So, relabeling the
left hand side by $\widehat{T}$, gives $T=\widehat{T}\in\mathfrak{f}^{0}$.

By Definition 4.3(ii), we have $\kappa(T,z)=0$, i.e.
$[T,z]_{\mathfrak{f}}=[T,z],\,\forall z\in\mathfrak{f}$. Then, by exploiting
the semi-simplicity of $\operatorname{ad}_{T}$, we next determine the
remaining basis elements $\widehat{x}\in\mathfrak{f}^{i}$ with the leading
parts $x\in\mathfrak{a}^{\Phi_{\mathbb{U}}}_{i}$, i.e.
$\operatorname{gr}_{i}(\widehat{x})=x$.

We first consider $\widehat{x}\in\mathfrak{f}^{0}$. We claim that without loss
of generality, as it was for $\widehat{T}\in\mathfrak{f}^{0}$ above, we may
assume that $x\in\mathfrak{f}^{0}$ for all
$x\in\mathfrak{a}^{\Phi_{\mathbb{U}}}_{0}$. We let
$\widehat{x}=x+c_{x}\mathsf{Y}$. Then, for $\mathbb{U}$ that is not PR the
claim holds, since $\mathsf{Y}\in\mathfrak{f}^{0}$, and so
$x=\widehat{x}-c_{x}\mathsf{Y}\in\mathfrak{f}^{0}$. In order to give the
argument for the case when $\mathbb{U}$ is PR, we recall that
$[T,\mathsf{Y}]=\mathsf{Y}$ and $[T,x]=0$ for all
$x\in\mathfrak{a}_{0}^{\Phi_{\mathbb{U}}}$. So,
$[T,\widehat{x}]_{\mathfrak{f}}=[T,\widehat{x}]=c_{x}\mathsf{Y}\in\mathfrak{f}^{0}$.
Now, since $(\mathfrak{f},[\cdot,\cdot]_{\mathfrak{f}})$ is a Lie algebra and
$\mathsf{Y}\not\in\mathfrak{f}^{0}$, then the closure condition
$[T,\widehat{x}]\in\mathfrak{f}^{0}$ implies that $c_{x}=0$. So,
$x=\widehat{x}\in\mathfrak{f}^{0}$.

Next, we similarly consider $\widehat{x}\in\mathfrak{f}^{i}$ for $i<0$. Recall
that by Definition 4.1 for these cases we have
$\mathfrak{a}^{\Phi_{\mathbb{U}}}_{i}=\mathfrak{g}_{i}$, for
$\mathfrak{g}_{i}$ as was defined in §2.2. In view of Lemma 4.7, we fix
$\operatorname{ad}_{T}$-invariant subspaces
$\mathfrak{s}^{\perp}\subset\mathfrak{g}$ in Table 6 such that
$\mathfrak{g}=\mathfrak{s}\oplus\mathfrak{s}^{\perp}$, where
$\mathfrak{s}:=\mathfrak{a}^{\Phi_{\mathbb{U}}}$, and define the deformation
map $\mathfrak{D}:\mathfrak{s}\to\mathfrak{s}^{\perp}$ (Definition 4.6). Let
$E^{i,a}$ and $\mathsf{X}^{\ast}$ denote the dual basis elements to $E_{i,a}$
and $\mathsf{X}$ respectively, and recall bi-grades for the basis elements
from Figure 1. Since, for $0\leq i\leq n,\,1\leq a,b,c\leq m$, the eigenvalues
of $\operatorname{ad}_{T}$ on

(5.3)
$\displaystyle\begin{array}[]{cccccc}E^{i,a}\otimes\mathsf{Z}_{1},&E^{i,a}\otimes
e_{c}{}^{b},&E^{i,a}\otimes\mathsf{Y},&\mathsf{X}^{\ast}\otimes\mathsf{Z}_{1},&\mathsf{X}^{\ast}\otimes
e_{c}{}^{b},&\mathsf{X}^{\ast}\otimes\mathsf{Y},\end{array}$

are $i-1,\,i-1,\,i,\,1,\,1,\,2\,$ respectively, then we have zero eigenvalues
only when $i=0$ or $1$. Then $T\cdot\mathfrak{D}=0$ (Lemma 4.7) implies
$\mathsf{X}=\widehat{\mathsf{X}}\in\mathfrak{f}$ and
$E_{i,a}=\widehat{E}_{i,a}\in\mathfrak{f}$ for all $i$ except possibly when
$i=0$ or $1$.

$\begin{array}[]{|cccc|}\hline\cr n&\mbox{Irreducible C-class module
}\mathbb{U}&\mbox{Generators
for}\,\mathfrak{s}^{\perp}\subset\mathfrak{g}&\mbox{Ranges}\\\
\hline\cr\hline\cr 2&\mathbb{B}_{4}&\mathsf{Z}_{1},\,\,e_{1}{}^{b}&2\leq b\leq
m\\\ \geq
3&\mathbb{A}_{2}^{\operatorname{tr}}&\mathsf{Z}_{1},\,\,e_{1}{}^{b},\,\,\mathsf{Y}&2\leq
b\leq m\\\
2&\mathbb{A}_{2}^{\operatorname{tf}}&\mathsf{Z}_{1},\,\,e_{1}{}^{b},\,\,e_{d}{}^{m}&2\leq
b\leq m,\,2\leq d\leq m-1\\\ \geq
3&\mathbb{A}_{2}^{\operatorname{tf}}&\mathsf{Z}_{1},\,\,e_{1}{}^{b},\,\,e_{d}{}^{m},\,\,\mathsf{Y}&2\leq
b\leq m,\,2\leq d\leq m-1\\\ \hline\cr\end{array}$ Table 6.
$\operatorname{ad}_{T}$-invariant subspace
$\mathfrak{s}^{\perp}\subset\mathfrak{g}$ complementary to
$\mathfrak{s}=\mathfrak{a}^{\Phi_{\mathbb{U}}}$

Now, consider the above exceptional cases. Based on the eigenvalues for
$\operatorname{ad}_{T}$ given in (5.3), we must have

(5.4) $\displaystyle\widehat{E}_{0,a}=E_{0,a}+\lambda_{a}\mathsf{Y}.$

Since $\mathbb{U}$ is a C-class module, then $\kappa(\mathsf{X},\cdot)=0$
(Definition 2.4), which implies
$[\mathsf{X},\cdot]_{\mathfrak{f}}=[\mathsf{X},\cdot]$. Recall that
$(\mathfrak{f},[\cdot,\cdot]_{\mathfrak{f}})$ is a Lie algebra and
$[\mathsf{X},E_{0,a}]=E_{1,a}$. Then for $\mathbb{U}$ that is

  * (a)

not PR ($\mathbb{U}=\mathbb{B}_{4}$ or $\mathbb{A}_{2}^{\operatorname{tf}}$
when $n=2$): we have that
$E_{0,a}=\widehat{E}_{0,a}-\lambda_{a}\mathsf{Y}\in\mathfrak{f}$, since
$\mathsf{Y}\in\mathfrak{f}$ and $\widehat{E}_{0,a}\in\mathfrak{f}$. Since
$\mathsf{X}\in\mathfrak{f}$, then
$[\mathsf{X},E_{0,a}]_{\mathfrak{f}}=[\mathsf{X},E_{0,a}]=E_{1,a}\in\mathfrak{f}$.
Hence, $E_{1,a}=\widehat{E}_{1,a}$ and so
$\mathfrak{f}=\mathfrak{a}^{\Phi_{\mathbb{U}}}$.

  * (b)

PR ($\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tf}}$ or
$\mathbb{A}_{2}^{\operatorname{tr}}$ when $n\geq 3$): Recall from our
discussion above that for any $\widehat{x}\in\mathfrak{f}^{0}$ with the
leading part $x\in\mathfrak{a}^{\Phi_{\mathbb{U}}}_{0}$, we may assume without
loss of generality that $x\in\mathfrak{f}^{0}$. Hence, Table 3 yields:

(5.5)
$\displaystyle\begin{split}q&:=e_{1}{}^{1}-e_{2}{}^{2}+\mathsf{Z}_{2}\in\mathfrak{f}^{0}\quad\mbox{for}\quad\mathbb{A}_{2}^{\operatorname{tr}}\\\
p&:=e_{1}{}^{1}-e_{2}{}^{2}+(2+\delta_{m-1}{}^{1})\mathsf{Z}_{2}\in\mathfrak{f}^{0}\,\quad\mbox{for}\quad\mathbb{A}_{2}^{\operatorname{tf}}\end{split}$

Recall that by Definition 4.3 (ii) we have $\kappa(\mathfrak{f}^{0},\cdot)=0$,
which implies $[z,\cdot]_{\mathfrak{f}}=[z,\cdot]$ for all
$z\in\mathfrak{f}^{0}$. Now, since both $p$ and $q$ commute with $\mathsf{Y}$,
$[\mathsf{Z}_{2},E_{i,a}]=-E_{i,a}$ and
$[e_{1}{}^{1}-e_{2}{}^{2},E_{i,a}]=(\delta_{a}{}^{1}-\delta_{a}{}^{2})E_{i,a}$
(see §2.2), then for

    * (i)

$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tf}}$: we have
$[p,\widehat{E}_{0,a}]_{\mathfrak{f}}=[p,\widehat{E}_{0,a}]=(\delta_{a}{}^{1}-\delta_{a}{}^{2}-\delta_{m-1}{}^{1}-2)E_{0,a}$.
So, the closure condition $[p,\widehat{E}_{0,a}]\in\mathfrak{f}$ forces
$\lambda_{a}=0$, i.e. $E_{0,a}=\widehat{E}_{0,a}\in\mathfrak{f}$. Then
$[\mathsf{X},E_{0,a}]=E_{1,a}\in\mathfrak{f}$, which implies
$E_{1,a}=\widehat{E}_{1,a}\in\mathfrak{f}$. Hence,
$\mathfrak{f}=\mathfrak{a}^{\Phi_{\mathbb{A}_{2}^{\operatorname{tf}}}}$. This
completes the proof for (a).

    * (ii)

$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tr}}$: we have
$[q,\widehat{E}_{0,a}]_{\mathfrak{f}}=[q,\widehat{E}_{0,a}]=(\delta^{1}_{a}-\delta^{2}_{a}-1)E_{0,a}$.
So, $[q,\widehat{E}_{0,a}]\in\mathfrak{f}$ implies that $\lambda_{a}=0$ except
for $a=1$, so for these cases we have
$E_{0,a}=\widehat{E}_{0,a}\in\mathfrak{f}$. Consequently,
$[\mathsf{X},E_{0,a}]=E_{1,a}\in\mathfrak{f}$ implies that
$E_{1,a}=\widehat{E}_{1,a}\in\mathfrak{f}$ except for $a=1$.

Finally, we consider the case when $a=1$. Based on the eigenvalues for
$\operatorname{ad}_{T}$ given in (5.3) and setting $\lambda_{1}=\zeta$, we
necessarily have

(5.6)
$\displaystyle\widehat{E}_{0,1}=E_{0,1}+\zeta\mathsf{Y}\quad\mbox{and}\quad\widehat{E}_{1,1}=E_{1,1}+\beta\mathsf{Z}_{1}+\sum_{b=2}^{m}\alpha_{b}e_{1}{}^{b}.$

Since from (2.8) we have $\mathsf{H}=n\mathsf{Z}_{2}-2\mathsf{Z}_{1}$ and
$T=\mathsf{Z}_{1}-\mathsf{Z}_{2}$, then we have

(5.7)
$\displaystyle\begin{split}[\mathsf{X},\widehat{E}_{0,1}]_{\mathfrak{f}}&=[\mathsf{X},E_{0,1}+\zeta\mathsf{Y}]=E_{1,1}+\zeta\mathsf{H}=E_{1,1}+\zeta(n\mathsf{Z}_{2}-2\mathsf{Z}_{1}),\\\
&=E_{1,1}+\zeta(n-2)\mathsf{Z}_{1}-\zeta nT.\end{split}$

So, the closure condition
$[\mathsf{X},\widehat{E}_{0,1}]_{\mathfrak{f}}\in\mathfrak{f}$ holds only if
$\widehat{E}_{1,1}=E_{1,1}+\zeta(n-2)\mathsf{Z}_{1}-\zeta nT$. This implies
$\beta=(n-2)\zeta$ and $\alpha_{b}=0$ for all $b$, which proves (b) and
concludes our proof.

∎

We have the following result for the curvature $\kappa$ of such an algebraic
model. This result is essential for our study of curvatures in §6.

######  Corollary 5.2.

Fix an irreducible C-class module
$\mathbb{U}\subset\mathbb{E}_{C}\subsetneq\mathbb{E}$, viewed as a
$G_{0}$-submodule of $\ker\square\subset C^{2}(\mathfrak{g}_{-},\mathfrak{g})$
via (2.18), and consider an algebraic model
$(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type with
$\kappa_{H}=\Phi_{\mathbb{U}}\in\mathbb{U}$, for $\Phi_{\mathbb{U}}$ from
Table 4, and $\dim\mathfrak{f}=\mathfrak{S}_{\mathbb{U}}$, normalized
according to Proposition 5.1. Then $\mathsf{X}\cdot\kappa=0$.

###### Proof.

Since $\mathbb{U}$ is a C-class module, then,
$\kappa\in\bigwedge^{2}V^{*}\otimes\mathfrak{g}$ (Remark 2.5) for $V$ from
§2.2. So, for $\mathsf{X}\in\mathfrak{f}$ we have
$[\mathsf{X},z]_{\mathfrak{f}}=[\mathsf{X},z]$ for all $z\in\mathfrak{f}$.
Then, as a consequence of the Jacobi identity we get the claim as follows:

(5.8)
$\displaystyle\begin{split}(\mathsf{X}\cdot\kappa)(x,y)&=[\mathsf{X},\kappa(x,y)]-\kappa([\mathsf{X},x],y)-\kappa(x,[\mathsf{X},y])\\\
&=[\mathsf{X},[x,y]]-\underbrace{[\mathsf{X},[x,y]_{\mathfrak{f}}]}_{[\mathsf{X},[x,y]_{\mathfrak{f}}]_{\mathfrak{f}}}+[\underbrace{[\mathsf{X},x]}_{[\mathsf{X},x]_{\mathfrak{f}}},y]_{\mathfrak{f}}-[[\mathsf{X},x],y]+[x,\underbrace{[\mathsf{X},y]}_{[\mathsf{X},y]_{\mathfrak{f}}}]_{\mathfrak{f}}-[x,[\mathsf{X},y]]=0.\end{split}$

∎

##  6\. Classification of submaximally symmetric vector ODEs of C-class

In this section, we classify (up to the $P$-action) all algebraic models of
ODE type for submaximally symmetric vector ODEs (1.2) of C-class (see the
introduction to §5) and establish Theorems 1.1, and 1.2.

###  6.1. Algebraic curvature constraints

For the algebraic models $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ whose
possible filtered linear subspaces $\mathfrak{f}\subset\mathfrak{g}$ have been
classified in Proposition 5.1, we classify their possible curvatures $\kappa$
below.

######  Proposition 6.1.

Fix an irreducible C-class module
$\mathbb{U}=\mathbb{B}_{4},\,\mathbb{A}_{2}^{\operatorname{tf}}$ or
$\mathbb{A}_{2}^{\operatorname{tr}}$ in $\mathbb{E}_{C}\subsetneq\mathbb{E}$,
viewed as a $G_{0}$-submodule of $\ker\square\subset
C^{2}(\mathfrak{g}_{-},\mathfrak{g})$ via (2.18), and consider an algebraic
model $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type with
$\kappa_{H}=\Phi_{\mathbb{U}}\in\mathbb{U}$, for $\Phi_{\mathbb{U}}$ from
Table 4, and $\dim\mathfrak{f}=\mathfrak{S}_{\mathbb{U}}$, normalized
according to Proposition 5.1. Then $\kappa$ is

  * (a)

$\mathbb{U}=\mathbb{B}_{4}:$ $\kappa=\pm\Phi_{\mathbb{U}}$ (over $\mathbb{C}$,
we can take $\kappa=\Phi_{\mathbb{U}}$);

  * (b)

$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tf}}:$ $\kappa=\Phi_{\mathbb{U}}$;

  * (c)

$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tr}}:$
$\kappa=\Phi_{\mathbb{U}}+\kappa_{4}$, where

(6.1) $\displaystyle\begin{split}\kappa_{4}&=\mu_{1}E^{3,1}\wedge
E^{0,1}\otimes\mathsf{X}+\mu_{2}E^{2,1}\wedge
E^{1,1}\otimes\mathsf{X}-\frac{\mu_{1}+\mu_{2}}{2}\left(E^{2,1}\wedge
E^{0,1}\otimes\mathsf{H}+E^{1,1}\wedge E^{0,1}\otimes\mathsf{Y}\right)\\\
&\quad+\mu_{3}\sum_{a=1}^{m}\left(E^{2,1}\wedge E^{0,a}-E^{2,a}\wedge
E^{0,1}+E^{1,a}\wedge E^{1,1}\right)\otimes e_{a}{}^{1},\end{split}$

for some $\mu_{1},\mu_{2},\mu_{3}\in\mathbb{R}$.

###### Proof.

The majority of the proof will consist of evaluating the annihilation
conditions $\mathfrak{f}^{0}\cdot\kappa=0$.

Recall from Table 2 that $\mathbb{U}$ has bi-grade either $(1,1)$ or $(2,2)$,
so $\mathsf{Z}_{1}-\mathsf{Z}_{2}\in\mathfrak{ann}(\Phi_{\mathbb{U}})$, which
is contained in $\mathfrak{f}^{0}$ by Proposition 5.1. Hence,
$(\mathsf{Z}_{1}-\mathsf{Z}_{2})\cdot\kappa=0$ implies that $\kappa$ is the
sum of terms with bi-grades that are multiples of $(1,1)$. But $\kappa$ is
regular, lies in
$\bigwedge^{2}(\mathfrak{g}/\mathfrak{p})^{\ast}\otimes\mathfrak{g}$, and
$\mathsf{Z}_{2}$ acts on the latter with eigenvalues $0,1$ or $2$. Thus, the
terms in $\kappa$ can only have bi-grades $(1,1)$ or $(2,2)$. By Theorem A.3,
$\kappa_{H}$ can be identified with the lowest $\mathsf{Z}$-degree component
of $\kappa$. Moreover, $\kappa_{H}$ is a nonzero multiple of
$\Phi_{\mathbb{U}}$. Using the $G_{0}$-action by $\exp(\mathsf{Z}t)$, where
$\mathsf{Z}\in\mathfrak{z}(\mathfrak{g}_{0})$ is the grading element (§2.2),
this multiple can be re-scaled to $\pm 1$. For
$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tf}}$ or
$\mathbb{A}_{2}^{\operatorname{tr}}$, we can further normalize this multiple
to $+1$. (Use the diagonal elements in
$g=\operatorname{diag}(a_{1},...,a_{m})\in\operatorname{GL}_{m}\subset G_{0}$,
i.e. $g\cdot\Phi^{i,j}=\frac{1}{a_{1}}\Phi^{i,j}$ for $\Phi^{i,j}$ from (3.5),
while $g\cdot\Phi^{i,j}=\frac{a_{m}}{(a_{1})^{2}}\Phi^{i,j}$ for $\Phi^{i,j}$
from (3.53).) Summarizing, we have

(6.2)
$\displaystyle\kappa=\begin{cases}\pm\Phi_{\mathbb{U}},&\mbox{when}\quad\mathbb{U}=\mathbb{B}_{4},\\\
\Phi_{\mathbb{U}}+\kappa_{4},&\mbox{when}\quad\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tf}}\quad\mbox{or}\quad\mathbb{A}_{2}^{\operatorname{tr}}.\end{cases}$

where $\kappa_{4}$ is the bi-grade $(2,2)$ component of $\kappa$. The
$\mathbb{B}_{4}$ case is complete, and we turn to the remaining cases.

Since $\mathbb{U}$ is a C-class module, then by Remark 2.5 we have
$\kappa\in\bigwedge^{2}V^{*}\otimes\mathfrak{g}$ in the notation of §2.2.
Recall $\mathfrak{g}=\mathfrak{q}\ltimes V$, and $\mathfrak{q}$ and $V$ have
$\mathsf{Z}_{2}$-degrees $0$ and $-1$ respectively (Figure 1). In particular,
$\Phi_{\mathbb{U}}\in\bigwedge^{2}V^{\ast}\otimes V$ and
$\kappa_{4}\in\bigwedge^{2}V^{\ast}\otimes\mathfrak{q}$. More precisely, since
$\kappa_{4}$ has bi-grade $(2,2)$, then in terms of the dual basis elements
$E^{i,a}$ to $E_{i,a}$, having bi-grades $(i,1)$ and $(-i,-1)$ respectively,
$\kappa_{4}$ must lie in the subspace
$K_{4}\subset\bigwedge^{2}V^{\ast}\otimes\mathfrak{q}$ spanned by

(6.3) $\displaystyle\begin{split}&E^{1,a}\wedge E^{0,b}\otimes\mathsf{Y},\quad
E^{3,a}\wedge E^{0,b}\otimes\mathsf{X},\quad E^{2,a}\wedge
E^{1,b}\otimes\mathsf{X},\\\ &E^{2,a}\wedge E^{0,b}\otimes\mathsf{H},\quad
E^{1,a}\wedge E^{1,b}\otimes\mathsf{H},\quad E^{2,a}\wedge E^{0,b}\otimes
e_{c}{}^{d},\quad E^{1,a}\wedge E^{1,b}\otimes e_{c}{}^{d},\end{split}$

where $1\leq a,b,c,d\leq m$. We will further constrain $\kappa_{4}$ as
follows. Using Proposition 5.1, we have
$\mathfrak{ann}(\Phi_{\mathbb{U}})\subset\mathfrak{f}^{0}$. Such elements
annihilate both $\Phi_{\mathbb{U}}$ and $\kappa$, and so

(6.4) $\displaystyle z\cdot\kappa_{4}=0,\quad\forall
z\in\mathfrak{ann}(\Phi_{\mathbb{U}}).$

Let us use these to find more explicit conditions on $\kappa_{4}$.

  * (1)

$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tf}}$: from Table 3, we have
$p:=e_{1}{}^{1}-e_{2}{}^{2}+(2+\delta_{m-1}{}^{1})\mathsf{Z}_{2}\in\mathfrak{ann}(\Phi_{\mathbb{U}})$.
From $0=p\cdot\kappa_{4}$ and $\mathsf{Z}_{2}\cdot\kappa_{4}=2\kappa_{4}$, we
have that $\kappa_{4}$ has eigenvalue $\lambda=-2(2+\delta_{m-1}{}^{1})$ for
$e_{1}{}^{1}-e_{2}{}^{2}$. We conclude that $\kappa_{4}=0$ (hence
$\kappa=\Phi_{\mathbb{U}}$) from the following considerations:

    1. (i)

$m=2$: We have $\lambda=-6$. Noting that $e_{1}{}^{1}-e_{2}{}^{2}$ commutes
with $\\{\mathsf{X},\mathsf{H},\mathsf{Y}\\}$, and

(6.5) $\displaystyle(e_{1}{}^{1}-e_{2}{}^{2})\cdot E^{i,a}$
$\displaystyle=(\delta_{a}{}^{2}-\delta_{a}{}^{1})E^{i,a},$ (6.6)
$\displaystyle(e_{1}{}^{1}-e_{2}{}^{2})\cdot e_{a}{}^{b}$
$\displaystyle=\delta_{a}{}^{1}e_{1}{}^{b}-\delta_{1}{}^{b}e_{a}{}^{1}-\delta_{a}{}^{2}e_{2}{}^{b}+\delta_{2}{}^{b}e_{a}{}^{2}.$

From (6.3), we conclude that the eigenvalues of $e_{1}{}^{1}-e_{2}{}^{2}$ in
$K_{4}$ lie between $-4$ and $4$. Since $-6$ is not an eigenvalue, then
$\kappa_{4}=0$.

    2. (ii)

$m\geq 3$: We have $\lambda=-4$. Proceeding as in (a), we observe that $K_{4}$
has $-4$-eigenspace for $e_{1}{}^{1}-e_{2}{}^{2}$ spanned by $E^{2,1}\wedge
E^{0,1}\otimes e_{2}{}^{1}$. But from Table 3, we also have
$e_{m-1}{}^{m-1}-e_{m}{}^{m}+\mathsf{Z}_{2}\in\mathfrak{ann}(\Phi_{\mathbb{U}})$,
which must similarly annihilate $\kappa$ and $\kappa_{4}$. But its eigenvalue
on $E^{2,1}\wedge E^{0,1}\otimes e_{2}{}^{1}$ is $\delta_{m-1}{}^{2}+2$, which
is nonzero, so $\kappa_{4}=0$ follows.

  * (2)

$\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tr}}$: from Table 3, we have
$q_{d}:=e_{d}{}^{d}-e_{d+1}{}^{d+1}+\delta_{1}{}^{d}\mathsf{Z}_{2}\in\mathfrak{ann}(\Phi_{\mathbb{U}})$,
so $0=q_{d}\cdot\kappa_{4}$ for $1\leq d\leq m-1$. Letting
$\mathfrak{h}\subset\mathfrak{sl}_{m}$ denote the standard Cartan subalgebra
consisting of diagonal trace-free matrices, and
$\epsilon_{a}\in\mathfrak{h}^{*}$ the standard weights for $\mathfrak{h}$, we
observe:

    1. (i)

$0=q_{d}\cdot\kappa_{4}$ for $1\leq d\leq m-1$ is equivalent to $\kappa_{4}$
having weight $-2\epsilon_{1}$;

    2. (ii)

the first five elements of (6.3) have weight $-\epsilon_{a}-\epsilon_{b}$.

    3. (iii)

the last two elements of (6.3) have weight
$-\epsilon_{a}-\epsilon_{b}+\epsilon_{c}-\epsilon_{d}$.

Matching these weights with $-2\epsilon_{1}$, we deduce that $\kappa_{4}$ lies
in the span of the following:

(6.7) $\displaystyle\begin{split}&E^{3,1}\wedge E^{0,1}\otimes\mathsf{X},\quad
E^{2,1}\wedge E^{1,1}\otimes\mathsf{X},\quad E^{2,1}\wedge
E^{0,1}\otimes\mathsf{H},\quad E^{1,1}\wedge E^{0,1}\otimes\mathsf{Y},\\\
&E^{2,1}\wedge E^{0,1}\otimes e_{1}{}^{1},\quad E^{2,1}\wedge E^{0,1}\otimes
e_{a}{}^{a},\quad E^{2,1}\wedge E^{0,a}\otimes e_{a}{}^{1},\quad E^{2,a}\wedge
E^{0,1}\otimes e_{a}{}^{1},\\\ &E^{1,a}\wedge E^{1,1}\otimes
e_{a}{}^{1},\end{split}$

where $2\leq a\leq m$. Similarly as in §3.1, we conclude that imposing
annihilation by all of
$\mathfrak{ann}(\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}})\subset\mathfrak{f}^{0}$
forces $\kappa_{4}$ to lie in the subspace spanned by

(6.8) $\displaystyle\begin{split}&E^{3,1}\wedge E^{0,1}\otimes\mathsf{X},\quad
E^{2,1}\wedge E^{1,1}\otimes\mathsf{X},\quad E^{2,1}\wedge
E^{0,1}\otimes\mathsf{H},\quad E^{1,1}\wedge E^{0,1}\otimes\mathsf{Y},\\\
&\sum_{a=1}^{m}E^{2,1}\wedge E^{0,1}\otimes
e_{a}{}^{a},\quad\sum_{a=1}^{m}E^{2,1}\wedge E^{0,a}\otimes
e_{a}{}^{1},\quad\sum_{a=1}^{m}E^{2,a}\wedge E^{0,1}\otimes
e_{a}{}^{1},\quad\sum_{a=1}^{m}E^{1,a}\wedge E^{1,1}\otimes
e_{a}{}^{1}.\end{split}$

Finally, we complete the proof by imposing $\mathsf{X}\cdot\kappa=0$
(Corollary 5.2). Since
$\mathsf{X}\cdot\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}=0$ (Lemma 2.2), then
$\mathsf{X}\cdot\kappa=0$ implies that $\mathsf{X}\cdot\kappa_{4}=0$. Now let
$\kappa_{4}$ be a general linear combination of all elements of (6.8), i.e.
$\kappa_{4}=\nu_{1}E^{3,1}\wedge E^{0,1}\otimes\mathsf{X}+\nu_{2}E^{2,1}\wedge
E^{1,1}\otimes\mathsf{X}+...+\nu_{8}\sum_{a=1}^{m}E^{1,a}\wedge E^{1,1}\otimes
e_{a}{}^{1}$, and impose $0=\mathsf{X}\cdot\kappa_{4}$ using the actions given
in §2.2. Namely, $\mathsf{X}\cdot\mathsf{Y}=\mathsf{H}$,
$\mathsf{X}\cdot\mathsf{H}=-2\mathsf{X}$, and $\mathsf{X}\cdot e_{a}{}^{b}=0$.
Also, $\mathsf{X}\cdot E_{i,a}=E_{i+1,a}$, and so $\mathsf{X}\cdot
E^{i,a}=-E^{i-1,a}$. We find that $0=\mathsf{X}\cdot\kappa_{4}$ is equivalent
to:

(6.9)
$\displaystyle\nu_{3}=\nu_{4}=-\frac{\nu_{1}+\nu_{2}}{2},\quad\nu_{5}=0,\quad\nu_{6}=\nu_{8}=-\nu_{7}.$

Setting $(\nu_{1},\nu_{2},\nu_{8})=(\mu_{1},\mu_{2},\mu_{3})$ then yields the
result.

∎

######  Corollary 6.2.

All parameters involved in an algebraic model
$(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type from Proposition 6.1
for $\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tr}}$ are uniquely determined.

###### Proof.

Recall from Table 2 that $\mathbb{U}=\mathbb{A}_{2}^{\operatorname{tr}}$
arises for $n\geq 3$. By Proposition 5.1 (b) and Proposition 6.1 (c), any
algebraic model $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type with
$\kappa_{H}=\Phi_{\mathbb{U}}\in\mathbb{U}$, for $\Phi_{\mathbb{U}}$ from
Table 4, and $\dim\mathfrak{f}=\mathfrak{S}_{\mathbb{U}}$ has

(6.10)
$\displaystyle\begin{split}&\mathfrak{f}=\operatorname{span}\\{E_{n,a},\ldots,E_{2,a},\widehat{E}_{1,1},E_{1,b},\widehat{E}_{0,1},E_{0,b},\mathsf{X}\,:\,1\leq
a\leq m,\,2\leq b\leq m\\}\oplus\mathfrak{ann}(\Phi_{\mathbb{U}}),\end{split}$

where $\mathfrak{ann}(\Phi_{\mathbb{U}})$ was given in Table 3, and

(6.11)
$\displaystyle\widehat{E}_{1,1}:=E_{1,1}+(n-2)\zeta\mathsf{Z}_{1}\in\mathfrak{f},\qquad\widehat{E}_{0,1}:=E_{0,1}+\zeta\mathsf{Y}\in\mathfrak{f}$

for some $\zeta\in\mathbb{R}$. Curvature is
$\kappa=\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}+\kappa_{4}$, for
$\kappa_{4}$ given in (6.1), and
$[\cdot,\cdot]_{\mathfrak{f}}=[\cdot,\cdot]-\kappa(\cdot,\cdot)$.

Let us now impose the Jacobi identity. We define

(6.12)
$\displaystyle\operatorname{Jac}^{\mathfrak{f}}(x,y,z):=[x,[y,z]_{\mathfrak{f}}]_{\mathfrak{f}}-[[x,y]_{\mathfrak{f}},z]_{\mathfrak{f}}-[y,[x,z]_{\mathfrak{f}}]_{\mathfrak{f}},\qquad\forall
x,y,z\in\mathfrak{f}.$

We calculate

(6.13)
$\displaystyle\begin{split}[\widehat{E}_{1,1},[E_{0,2},E_{3,1}]_{\mathfrak{f}}]_{\mathfrak{f}}&=-(n-2)^{2}\left(2\zeta+\frac{3(2m+3)-n(4m+3)}{mn(n+1)+6}\right)E_{2,2},\\\
[E_{0,2},[\widehat{E}_{1,1},E_{3,1}]_{\mathfrak{f}}]_{\mathfrak{f}}&=-(n-2)^{2}\left(3\zeta+\frac{3(3m+5)-n(5m+3)}{mn(n+1)+6}\right)E_{2,2},\\\
[[\widehat{E}_{1,1},E_{0,2}]_{\mathfrak{f}},E_{3,1}]_{\mathfrak{f}}&=-\frac{(n-1)(n-2)^{2}(mn-3)}{mn(n+1)+6}E_{2,2},\\\
\end{split}$

so that

(6.14)
$\displaystyle\operatorname{Jac}^{\mathfrak{f}}(\widehat{E}_{1,1},E_{0,2},E_{3,1})=0\quad\mbox{implies}\quad\zeta=\frac{(2n-n^{2}-3)m+3n-9}{mn(n+1)+6}.$

Continuing in a similar manner, we find that:

(6.15)
$\displaystyle\begin{split}&\operatorname{Jac}^{\mathfrak{f}}(\widehat{E}_{0,1},E_{1,2},E_{3,1})=0\quad\mbox{
implies}\quad\mu_{1}=\frac{6(n-1)(n-2)(m+1)}{mn(n+1)+6},\\\
&\operatorname{Jac}^{\mathfrak{f}}(\widehat{E}_{1,1},E_{2,2},E_{2,1})=0\quad\mbox{
implies}\quad\mu_{2}=-\frac{6(n-1)(m+1)(m(n^{3}+n^{2}-6n+6)+6)}{(mn(n+1)+6)^{2}}.\end{split}$

Using $\zeta$ and $\mu_{1}$ above, we then have

(6.16)
$\displaystyle\operatorname{Jac}^{\mathfrak{f}}(\widehat{E}_{0,1},E_{2,2},E_{3,1})=0\quad\mbox{implies}\quad\mu_{3}=1-n.$

As claimed, the parameters $\zeta,\mu_{1},\mu_{2},\mu_{3}$ are uniquely
determined functions of $(n,m)$.

We remark that the remaining Jacobi identities for $\mathfrak{f}$ are
necessarily satisfied because the existence of a submaximally symmetric ODE
model in the $\mathbb{A}_{2}^{\operatorname{tr}}$-branch (see Table 1)
guarantees the existence of a corresponding algebraic model of ODE type.
(Necessarily, this is equivalent to the one found above.) ∎

###  6.2. Conclusion

Fix an irreducible C-class module
$\mathbb{U}=\mathbb{B}_{4},\mathbb{A}_{2}^{\operatorname{tr}}$ or
$\mathbb{A}_{2}^{\operatorname{tf}}$ in the effective part $\mathbb{E}$, and
recall the respective lowest weight vectors $\Phi_{\mathbb{U}}\in\mathbb{U}$
from Table 4. By Propositions 5.1 and 6.1, the classification of algebraic
models $(\mathfrak{f};\mathfrak{g},\mathfrak{p})$ of ODE type with
$0\not\equiv\operatorname{im}(\kappa_{H})\subset\mathbb{U}$ and
$\dim\mathfrak{f}=\mathfrak{S}_{\mathbb{U}}$ is given in Table 7.

$\begin{array}[]{|c|c|c|c|}\hline\cr n&\begin{array}[]{c}\mbox{Irreducible
C-class}\\\
\mbox{module}\,\,\mathbb{U}\subset\mathbb{E}\end{array}&\mathfrak{f}&\kappa\\\
\hline\cr\hline\cr
2&\mathbb{B}_{4}&\mathfrak{a}^{\Phi_{\mathbb{U}}}&\begin{cases}\Phi_{\mathbb{U}},&\mbox{over}\,\,\mathbb{C}\\\
\pm\Phi_{\mathbb{U}},&\mbox{over}\,\,\mathbb{R}\end{cases}\\\ \hline\cr\geq
3&\mathbb{A}_{2}^{\operatorname{tr}}&\begin{array}[]{l}\mathfrak{f}\,\,\mbox{in}\,\,\eqref{E:fA2}\,\,\mbox{with}\\\
\,\zeta\,\,\mbox{in}\,\,\eqref{E:zeta}\end{array}&\begin{array}[]{l}\Phi_{\mathbb{U}}+\kappa_{4},\,\mbox{with}\,\beta=1,\kappa_{4}\,\,\mbox{in}\,\eqref{E:kappa4},\\\
\mbox{and}\,\mu_{1},\mu_{2},\mu_{3}\,\,\mbox{in}\,\,\eqref{E:mu}\,\,\mbox{and}\,\,\eqref{E:mu3}.\end{array}\\\
\hline\cr\geq
2&\mathbb{A}_{2}^{\operatorname{tf}}&\mathfrak{a}^{\Phi_{\mathbb{U}}}&\Phi_{\mathbb{U}}\\\
\hline\cr\end{array}$ Table 7. Classification of algebraic models of ODE type
with $0\not\equiv\operatorname{im}(\kappa_{H})\subset\mathbb{U}$ and
$\dim\mathfrak{f}=\mathfrak{S}_{\mathbb{U}}$

We now discuss how the ODE model classification in Table 1 is deduced from the
abstract classification in Table 7. Using fundamental invariants described in
§2.4, we confirm that these ODE lie in the claimed branches. In [13, Table
10], the point symmetries were given for all of these models with the
exception of the second $\mathbb{B}_{4}$ model. (See below for this case.) We
confirm submaximal symmetry and deduce the associated algebraic models. (This
is immediate by uniqueness in the
$\mathbb{A}_{2}^{\operatorname{tr}},\mathbb{A}_{2}^{\operatorname{tf}}$ cases,
as well as the $\mathbb{B}_{4}$ case over $\mathbb{C}$.)

To complete the proof of Theorem 1.2, we establish point-inequivalence over
$\mathbb{R}$ of the following $\mathbb{B}_{4}$ models:

(6.17) $\displaystyle
u_{3}^{a}=\displaystyle\frac{3u_{2}^{1}u_{2}^{a}}{2u^{1}_{1}}\qquad\mbox{or}\qquad
u_{3}^{a}=\displaystyle\frac{3u_{1}^{1}u_{2}^{1}u_{2}^{a}}{1+(u^{1}_{1})^{2}}\qquad\mbox{for}\quad
1\leq a\leq m.$

The point symmetries of the former are given in [13, Table 10], from which we
observe that the distribution $\ker(du^{1})$ on
$J^{0}(\mathbb{R},\mathbb{R}^{m})$ is invariant under the action of the
symmetry algebra $\mathcal{S}$. This implies that the foliation in
$J^{0}(\mathbb{R},\mathbb{R}^{m})$ by level sets of $u^{1}$ is also
$\mathcal{S}$-invariant, and total differentiation implies that
$\\{u_{1}^{1}=0\\}\subset J^{1}(\mathbb{R},\mathbb{R}^{m})$ is
$\mathcal{S}$-invariant, i.e. the prolonged action of $\mathcal{S}$ on
$J^{1}(\mathbb{R},\mathbb{R}^{m})$ is not locally transitive.

In contrast, we now establish that the latter ODE in (6.17) has symmetry
algebra that acts locally transitively on $J^{1}(\mathbb{R},\mathbb{R}^{m})$.
Its point symmetries are (for $1\leq a\leq m$ and $2\leq b\leq m$):

(6.18) $\displaystyle\begin{split}&\partial_{t},\quad\partial_{u^{a}},\quad
t\partial_{u^{b}},\quad
u^{a}\partial_{u^{b}},\quad(t^{2}+(u^{1})^{2})\partial_{u^{b}},\quad
u^{1}\partial_{t}-t\partial_{u^{1}},\quad
t\partial_{t}+u^{1}\partial_{u^{1}}+2\sum_{b=2}^{m}u^{b}\partial_{u^{b}},\\\
&(t^{2}-(u^{1})^{2})\partial_{t}+2t\sum_{a=1}^{m}u^{a}\partial_{u^{a}},\quad
tu^{1}\partial_{t}+\frac{1}{2}((u^{1})^{2}-t^{2})\partial_{u^{1}}+u^{1}\sum_{b=2}^{m}u^{b}\partial_{u^{b}}.\end{split}$

In particular over $\mathbb{R}$, transitivity immediately follows from
prolonging some of them to $J^{1}(\mathbb{R},\mathbb{R}^{m})$:

(6.19) $\displaystyle\partial_{t},\quad\partial_{u^{a}},\quad
t\partial_{u^{b}}+\partial_{u^{b}_{1}},\quad
u^{1}\partial_{t}-t\partial_{u^{1}}-(1+(u^{1}_{1})^{2})\partial_{u^{1}_{1}}-u^{1}_{1}\sum_{b=2}^{m}u^{b}_{1}\partial_{u^{b}_{1}}.$

We conclude that the symmetry algebras of (6.17) are point-inequivalent, and
hence the ODEs are point-inequivalent. We also remark that when $m=1$, the two
3rd order ODEs (6.17) are point-inequivalent scalar ODEs with symmetry algebra
$\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)$ and $\mathfrak{so}(3,1)$ respectively
[1, p. 18].

This completes the proof of Theorem 1.2. We have also proven the remaining
Theorem 1.1(b) since for vector ODEs (1.2) of C-class of order $n+1\geq 3$, we
have
$\mathfrak{S}=\mathfrak{M}-2=\mathfrak{S}_{\mathbb{B}_{4}}=\mathfrak{S}_{\mathbb{A}_{2}^{\operatorname{tf}}}$
only when $(n,m)=(2,2)$. This completes our proofs for Theorems 1.1 and 1.2.

##  Appendix A Harmonic curvature as the lowest degree component of curvature

Fix $G$ and $P$ as in §2.2 and recall from §2.3 some basic notions of Cartan
geometries $(\mathcal{G}\to\mathcal{E},\omega)$ of type $(G,P)$ associated to
ODEs (1.2). We formulate Theorem A.3 below stating that the harmonic curvature
$\kappa_{H}$ can be identified with the lowest degree component (with respect
to the grading element) of the curvature $\kappa$. (We note that this is used
in the proof of Proposition 6.1, which is essential in proving Theorems 1.1
and 1.2.)

######  Definition A.1.

Let $(\mathcal{G}\to\mathcal{E},\omega)$ be a Cartan geometry of type $(G,P)$,
let $\rho:G\to\operatorname{GL}(V)$ be a $G$-representation, and
$\rho\circ\iota:P\to\operatorname{GL}(V)$ its restriction, where
$\iota:P\hookrightarrow G$ is the canonical inclusion. A tractor bundle is an
associated vector bundle $\mathcal{G}\times_{P}V$ with respect to the
$P$-representation $\rho\circ\iota$. Given the adjoint representation
$\rho=\operatorname{Ad}:G\to\operatorname{GL}(\mathfrak{g})$, the tractor
bundle $\mathcal{A}\mathcal{E}:=\mathcal{G}\times_{P}\mathfrak{g}$ is called
the adjoint tractor bundle (see [5, §1.5.7] for further details).

Using the Cartan connection $\omega$, the tangent bundle $T\mathcal{E}$ can be
identified with the bundle $\mathcal{G}\times_{P}(\mathfrak{g}/\mathfrak{p})$.
Then, the $P$-invariant quotient map from $\mathfrak{g}$ onto
$\mathfrak{g}/\mathfrak{p}$ gives rise to the natural projection
$\Pi:\mathcal{A}\mathcal{E}\to T\mathcal{E}$. Using this identification, we
can regard the curvature as
$\kappa\in\Omega^{2}(\mathcal{E},\mathcal{A}\mathcal{E})$, i.e.
$\mathcal{A}\mathcal{E}$-valued 2-form on $\mathcal{E}$ [5, Prop 1.5.7].

######  Definition A.2.

Given a Cartan geometry $(\mathcal{G}\to\mathcal{E},\omega)$ of type $(G,P)$
with curvature $\kappa\in\Omega^{2}(\mathcal{E},\mathcal{A}\mathcal{E})$.
Then:

  * (a)

$\omega$ is called regular if
$\kappa\in(\Omega^{2}(\mathcal{E},\mathcal{A}\mathcal{E}))^{1}$, i.e.
$\kappa(T^{i}\mathcal{E},T^{j}\mathcal{E})\subset\mathcal{A}^{i+j+1}\mathcal{E},\,\,\forall
i,j<0$.

  * (b)

$\omega$ is called normal if $\partial^{\ast}\kappa=0$.

  * (c)

If $\omega$ is both regular and normal, then the harmonic curvature is
$\kappa_{H}:=\kappa\mod\operatorname{im}(\partial^{*})$, which is a section of
$\mathcal{G}\times_{P}\frac{\ker\partial^{*}}{\operatorname{im}\partial^{*}}$.

Then, we have the following result.

######  Theorem A.3.

Fix $G$ and $P$ as in §2.2. Let $(\mathcal{G}\to\mathcal{E},\omega)$ be a
regular, normal Cartan geometry of type $(G,P)$ whose curvature
$\kappa\in\left(\Omega^{2}(\mathcal{E},\mathcal{A}\mathcal{E})\right)^{\ell}$
for some $\ell\geq 1$, i.e.
$\kappa\left(T^{i}\mathcal{E},T^{j}\mathcal{E}\right)\subset\mathcal{A}^{i+j+\ell}\mathcal{E}$
for all $i,j<0$. Then the induced section
$\operatorname{gr}_{\ell}(\kappa)\in\operatorname{gr}_{\ell}\left(\Omega^{2}(\mathcal{E},\mathcal{A}\mathcal{E})\right)$
coincides with the degree $\ell$ component of the harmonic curvature
$\kappa_{H}$. Consequently, $\kappa_{H}\equiv 0$ implies $\kappa\equiv 0$.

###### Proof.

The statement was proved in [5, Theorem 3.1.12] for parabolic geometries. The
same proof works for our non-parabolic Cartan geometries associated to vector
ODEs (1.2) of order $\geq 3$. ∎

##  Appendix B A necessary condition for coclosedness of
$\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}$

From §3.2, our strategy for computing a $\mathfrak{sl}(W)$-lowest weight
vector
$\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}\in\mathbb{A}_{2}^{\operatorname{tr}}$
involves imposing coclosedness, i.e.
$\partial^{*}\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}=0$, where
$\partial^{*}$ was defined in §2.3. By adjointness of $\partial$ and
$\partial^{*}$ with respect to the inner product $\langle\cdot,\cdot\rangle$
on cochains induced from Definition 2.1, we have:

(B.1)
$\displaystyle\partial^{\ast}\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}=0\quad\iff\quad\left\langle\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}},\partial\psi\right\rangle=0,\quad\forall\psi\in\mathfrak{g}^{*}_{-}\otimes\mathfrak{g}.$

In order to pin down $\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}$ in
Proposition 3.5, only a small part of the conditions in (B.1) will be in fact
required. In this section, we identify a key condition (Lemma B.2) that is
essential to the proof of Proposition 3.5.

Recalling $\mathfrak{ann}(\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}})$ given in
Table 3, let us restrict attention to $\psi$ lying in the subspace below.

######  Lemma B.1.

Suppose that $\psi\in\mathfrak{g}_{-}^{*}\otimes\mathfrak{g}$ has bi-grade
$(1,1)$, with $\mathsf{X}\cdot\psi=0$ and
$\mathfrak{ann}(\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}})\cdot\psi=0$. Then
$\psi$ is a multiple of

(B.2)
$\displaystyle\Psi:=-2E^{2,1}\otimes\mathsf{X}+E^{1,1}\otimes\mathsf{H}+E^{0,1}\otimes\mathsf{Y}.$

###### Proof.

Any $\psi\in\mathfrak{g}_{-}^{*}\otimes\mathfrak{g}$ with bi-grade $(1,1)$
lies in the span of:

(B.3) $\displaystyle E^{2,a}\otimes\mathsf{X},\quad
E^{1,a}\otimes\mathsf{H},\quad E^{0,a}\otimes\mathsf{Y},\quad E^{1,a}\otimes
e_{b}{}^{c}\quad(1\leq a,b,c\leq m).$

Since $\mathsf{X}\cdot E_{i,a}=E_{i+1,a}$, then $\mathsf{X}\cdot
E^{i,a}=-E^{i-1,a}$. Imposing $\mathsf{X}\cdot\psi=0$ forces $\psi$ to lie in
the span of:

(B.4)
$\displaystyle-2E^{2,a}\otimes\mathsf{X}+E^{1,a}\otimes\mathsf{H}+E^{0,a}\otimes\mathsf{Y}\quad(1\leq
a\leq m).$

Let us now impose
$\mathfrak{ann}(\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}})\cdot\psi=0$. Recall
from Table 3 that
$q_{d}=e_{d}{}^{d}-e_{d+1}{}^{d+1}+\delta_{1}{}^{d}\mathsf{Z}_{2}\in\mathfrak{ann}(\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}})$
for $1\leq d\leq m-1$. Let $\mathfrak{h}\subset\mathfrak{sl}_{m}$ denote the
standard Cartan subalgebra consisting of diagonal trace-free matrices, and
$\epsilon_{a}\in\mathfrak{h}^{*}$ the standard weights for $\mathfrak{h}$.
Since $\mathsf{Z}_{2}\cdot\psi=\psi$, then

(B.5) $\displaystyle q_{d}\cdot\psi=0\quad\mbox{for}\quad 1\leq d\leq
m-1\iff\psi\quad\mbox{has weight}\quad-\epsilon_{1}.$

Since each element of (B.4) has weight $-\epsilon_{a}$, then being of weight
$-\epsilon_{1}$ implies that $\psi$ is a multiple of (B.2). We note that (B.2)
is annihilated by all off-diagonal elements
$e_{f}{}^{d}\in\mathfrak{ann}(\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}})$,
since we have $f\geq 2$ and $e_{f}{}^{d}$ commutes with
$\\{\mathsf{X},\mathsf{H},\mathsf{Y}\\}$. This completes the proof. ∎

In terms of $\Phi^{i,j}=\sum_{a=1}^{m}E^{i,1}\wedge E^{j,a}\otimes
E_{i+j-1,a}$ defined in (3.5), and using $\partial$ (2.16), we get:

(B.6)
$\displaystyle\begin{split}\partial\Psi=-2\sum_{k=0}^{n-1}\Phi^{2,k}+\sum_{k=0}^{n}(2k-n)\Phi^{1,k}+\sum_{k=1}^{n}k(n+1-k)\Phi^{0,k}.\end{split}$

We then have the following necessary condition, which will be used in the
proof of Proposition 3.5.

######  Lemma B.2.

Take $\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}$ defined in Proposition 3.3,
i.e.
$\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}=\sum_{i,j=0}^{n}c_{i,j}\Phi^{i,j}$
with $c_{i,j}$ satisfying (3.6). Then

(B.7) $\displaystyle
0=\sum_{k=0}^{n-1}\frac{(n-k)(k+1)}{n(n-1)}(c_{k,2}-mc_{2,k})+\sum_{k=0}^{n}\frac{2k-n}{n}(mc_{1,k}-c_{k,1})+\sum_{k=1}^{n}(mc_{0,k}-c_{k,0}).$

###### Proof.

We evaluate (B.1) for $\psi=\Psi$ given in (B.2). In preparation for this,
note that from Definition 2.1, we have $\langle
E_{k,a},E_{k,a}\rangle=\frac{k!}{(n-k)!}$ and $\langle
E^{k,a},E^{k,a}\rangle=\frac{(n-k)!}{k!}$, and so

(B.8) $\displaystyle||E^{2,1}\wedge E^{k,a}\otimes
E_{k+1,a}||^{2}=||E^{2,1}||^{2}||E^{k,a}||^{2}||E_{k+1,a}||^{2}=\frac{(n-k)(k+1)(n-2)!}{2}.$

Hence, by bilinearity of $\langle\cdot,\cdot\rangle$ and orthogonality of the
basis elements for $\mathfrak{g}$ (Definition 2.1), we have

(B.9)
$\displaystyle\begin{split}\langle\Phi^{2,k},\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}\rangle&=\sum_{i,j=0}^{n}\sum_{a,b=1}^{m}c_{i,j}\left\langle
E^{2,1}\wedge E^{k,b}\otimes E_{k+1,b},E^{i,1}\wedge E^{j,a}\otimes
E_{i+j-1,a}\right\rangle\\\
&=\sum_{i,j=0}^{n}\sum_{a=1}^{m}(\delta_{i}{}^{2}\delta_{j}{}^{k}-\delta_{i}{}^{k}\delta_{j}{}^{2}\delta_{a}{}^{1})c_{i,j}||E^{2,1}\wedge
E^{k,a}\otimes E_{k+1,a}||^{2}\\\
&=\sum_{a=1}^{m}(c_{2,k}-c_{k,2}\delta_{a}{}^{1})\frac{(n-k)(k+1)(n-2)!}{2}\\\
&=(mc_{2,k}-c_{k,2})\frac{(n-k)(k+1)(n-2)!}{2}\end{split}$

Similarly, we have

(B.10)
$\displaystyle\langle\Phi^{1,k},\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}\rangle=(n-1)!(mc_{1,k}-c_{k,1}),\qquad\langle\Phi^{0,k},\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}\rangle=\frac{n!}{k(n+1-k)}(mc_{0,k}-c_{k,0}).$

We use these relations and (B.6) to evaluate
$0=\langle\partial\Psi,\Phi_{\mathbb{A}_{2}^{\operatorname{tr}}}\rangle$ and
obtain the claimed result. ∎

## Acknowledgements

The authors acknowledge the use of the DifferentialGeometry package in Maple.
We also acknowledge helpful conversations with Eivind Schneider and Andreu
Llabres. The research leading to these results has received funding from the
Norwegian Financial Mechanism 2014-2021 (project registration number
2019/34/H/ST1/00636), the Tromsø Research Foundation (project “Pure
Mathematics in Norway”), and the UiT Aurora project MASCOT.

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