Invariants of Surfaces in Three-Dimensional Affine Geometry

Invariants of Surfaces
in Three-Dimensional Affine Geometry

Örn ARNALDSSON a and Francis VALIQUETTE b

Ö. Arnaldsson and F. Valiquette

a) Department of Mathematics, University of Iceland, Reykjavik, Ssn.
600169-2039, Iceland<EMAIL_ADDRESS>
b) Department of Mathematics, Monmouth University, West Long Branch, NJ 07764,
USA<EMAIL_ADDRESS>
Received September 03, 2020, in final form March 21, 2021; Published online
March 30, 2021

Using the method of moving frames we analyze the algebra of differential
invariants for surfaces in three-dimensional affine geometry. For elliptic,
hyperbolic, and parabolic points, we show that if the algebra of differential
invariants is non-trivial, then it is generically generated by a single
invariant.

affine group; differential invariants; moving frames

22F05; 53A35; 53A55

##  1 Introduction

The local geometry of $p$-dimensional submanifolds $S$ of an $m$-dimensional
manifold $M$, under the smooth action of a Lie group $G$ is entirely governed
by their differential invariants, in the sense that two submanifolds are
locally congruent if and only if their differential invariants match [8, 9]. A
differential invariant is a (possibly locally defined) smooth function on the
submanifold jet bundle ${\rm J}^{(\infty)}={\rm J}^{\infty}(M,p)$ that remains
unchanged under the prolonged action of $G$. This prolonged action on ${\rm
J}^{(\infty)}$ splits/reduces to an action on $G$-invariant subbundles (called
_branches of the equivalence problem_) whose symmetry properties differ; some
branches having an infinite number of differential invariants of progressively
higher and higher order while others have no invariants. The _fundamental
basis theorem_ , first formulated in [16, p. 760], states that, on branches
with non-trivial invariants, all the differential invariants can be generated
from a finite number of low order invariants and their derivatives with
respect to $p$ invariant total derivative operators
$\mathcal{D}_{1},\dots,\mathcal{D}_{p}$. For example, differential invariants
of planar curves under the special Euclidean group ${\rm SE}(2)$ can all be
expressed in terms of the curvature and its (repeated) arc-length derivatives
[21]. We note that modern proofs of the fundamental basis theorem can be found
in [14, 15, 26] and that this theorem is also frequently called the Lie–Tresse
theorem.

A basic question, then, is to find a minimal generating set of invariants.
According to the above, such a set will completely determine the local
geometric properties of submanifolds under $G$. The equivariant moving frame
method is ideally suited for this type of question. Indeed, the effectiveness
of the equivariant moving frame method lies in its recurrence relations,
through which one obtains the complete and explicit structure of the
underlying algebra of differential invariants, and this without requiring
explicit coordinate expressions for the moving frame or the invariants,
leading to what is now referred to as the _symbolic invariant calculus_ [17].
In [11] and [23], this was applied to deduce the surprising result that there
is a single generating invariant for (suitably generic) surfaces in
$\mathbb{R}^{3}$ under the projective, conformal, Euclidean and equi-affine
groups. For the Euclidean and equi-affine groups, the algebra of differential
invariants is generically governed by the Gaussian curvature and Pick
invariant, respectively. Similarly, the algebra of differential invariants
under the equi-affine group for generic parabolic surfaces with nonvanishing
Pocchiola 4th invariant has recently been shown to be generated by a single
differential invariant in [3].

In the current paper we study the geometry of surfaces under the entire affine
group, ${\rm A}(3)={\rm GL}(3)\ltimes\mathbb{R}^{3}$, in detail. We do not
restrict ourselves to the most generic branch of surfaces as in [11, 23], but
rather provide all the different branches that have non-trivial invariants. In
each case, we study the algebra of differential invariants and obtain explicit
formulas, in terms of surface jets, for the generating invariants and the
invariants responsible for the various branchings. In certain cases, obtaining
expressions for the invariants using the direct moving frame approach proved
intractable. We therefore relied on the recently developed technique of
_recursive moving frames_ [24], to obtain the desired coordinate formulas. The
main result of our paper is that whenever a branch admits differential
invariants, the differential invariant algebra is (generically) generated by a
single invariant.

It is worth mentioning that, historically, differential geometers have given
more attention to the problem of finding and classifying _homogeneous spaces_
within a given equivalence problem of submanifolds $S\subset M$ under the
action of a Lie group $G$. We recall that homogeneous spaces are, by
definition, submanifolds that admit no non-trivial differential invariants,
and, in a sense, the study of these spaces is the “opposite” problem
considered in this paper as we focus our attention to surfaces that admit non-
trivial differential invariants. But for completeness, we note that the
classification of homogeneous surfaces in $\mathbb{R}^{3}$ under the equi-
affine group can be found in [10, Theorem 12.4] and [12, Chapter VI]. More
recently, normal forms for homogeneous surfaces in $\mathbb{R}^{3}$ under the
general affine group with vanishing equi-affine Pick invariant were found in
[1], and more generally in [5] and [7]. We note that since coordinate
expressions for all relative and differential invariants derived in this paper
are known, these could, theoretically, be used to find normal forms for the
homogeneous surfaces. In Section 6 we provide several examples and show that a
more efficient approach to deriving homogeneous surfaces is to integrate the
moving frame equations. Though we emphasize that the study of homogeneous
surfaces is not the main focus of the present paper.

We would be remiss if we failed to acknowledge the classical works of W.
Blaschke [2], and P. Schirokov and A. Schirokov [27] on the subject. Together
with [4], and the references therein, they provide a classical treatment of
affine differential geometry. The basic affine differential invariants can be
found in these classical works, and the main contribution of our paper is the
detailed analysis of the structure of the algebra of these differential
invariants for surfaces in affine 3-space.

We note that for parabolic surfaces, the problem studied in this paper is
related to the local geometry of 2-nondegenerate real analytic hypersurfaces
$S^{5}\subset\mathbb{C}^{3}$ in CR-geometry [18]. This correspondence is not
considered here, but we note that Question 7.1 in [18, Section 7] is solved in
this paper and corresponds to Case P.1.1 and its subcases. It is worth noting
that [18] has recently been superseded by the work of Doubrov, Merker, and The
in [6].

For a summary of the results obtained in this paper we refer the reader to
Section 7. As for the rest of the paper, in Section 2 we recall the notion of
a partial moving frame, introduce the recurrence relations that unlock the
structure of the algebra of differential invariants, summarize the recursive
moving frame implementation used to compute coordinate expressions of
invariants, and finally recall basic results pertaining to the algebra of
differential invariants. Sections 3, 4, and 5 contain the main results of this
paper. In Section 3 we initiate the normalization process up to order two. At
this order there is a splitting according to whether points are elliptic,
hyperbolic, or parabolic. In Section 4 we simultaneously consider elliptic and
hyperbolic points. Finally, in Section 5 we consider parabolic points.

##  2 Background material

In this section we recall basic results pertaining to the method of moving
frames. We refer the reader to the original manuscripts [8, 13, 24] and the
book [17] for a more comprehensive exposition.

###  2.1 Partial moving frames

In this section we introduce the notion of a partial moving frames as
introduced in [24]. Let $G$ be an $r$-dimensional Lie group acting on an
$m$-dimensional manifold $M$. We are interested with the induced action of $G$
on $p$-dimensional submanifolds $S\subset M$, where $1\leq p<m$ is fixed. For
$0\leq n\leq\infty$, let ${\rm J}^{(n)}={\rm J}^{(n)}(M,p)$ denote the $n$th
order submanifold jet bundle. Given the local coordinates
$z=(x,u)=\big{(}x^{1},\dots,x^{p},u^{1},\dots,u^{q}\big{)}$ on $M$, where $x$
are viewed as the independent variables and $u$ as the dependent variables,
coordinates on ${\rm J}^{(n)}$ are given by
$z^{(n)}=\big{(}x,u^{(n)}\big{)}=(\dots,x^{i},\dots,u^{\alpha}_{J},\dots)$,
where $u^{\alpha}_{J}$ denote the derivative coordinates of orders
$0\leq\\#J\leq n$.

Let $\mathcal{S}^{(n)}\subset{\rm J}^{(n)}$ be a $G$-invariant subbundle of
${\rm J}^{(n)}$ such that for all $g\in G$ near the identity,
$g\cdot\mathcal{S}^{(n)}\subseteq\mathcal{S}^{(n)}$. Such an invariant
subbundle is specified by a set of invariant differential equations

$\displaystyle\mathcal{S}^{(n)}=\big{\\{}z^{(n)}\in{\rm J}^{(n)}\mid
F\big{(}z^{(n)}\big{)}=0,\ \text{where}\ F\big{(}g\cdot
z^{(n)}\big{)}\big{|}_{F(z^{(n)})=0}=0\big{\\}}.$ (2.1)

The prolongation $\mathcal{S}^{(n+1)}$ is obtained by appending the
derivatives of the defining equations:

$\displaystyle\mathcal{S}^{(n+1)}=\big{\\{}z^{(n)}\in{\rm J}^{(n)}\mid
F\big{(}z^{(n)}\big{)}=0,\,(D_{1}F)\big{(}z^{(n+1)}\big{)}=0,\dots,(D_{p}F)\big{(}z^{(n+1)}\big{)}=0\big{\\}},$

where $D_{i}=D_{x^{i}}$ denote the total derivative operators. The induced
action of $G$ on $\mathcal{S}^{(n)}$ is called the _$n$ th order prolonged
action_. Borrowing Cartan’s notational convention, we use capital letters to
denote transformed variables: $Z^{(n)}=g\cdot z^{(n)}$. Let
$\mathcal{B}^{(n)}=G\times\mathcal{S}^{(n)}$ denote _$n$ th order lifted
bundle_. For $k\geq n$, we introduce the standard projection
$\pi^{k}_{n}\colon\mathcal{B}^{(k)}\to\mathcal{B}^{(n)}$. The lifted bundle
admits a groupoid structure with source map
$\sigma^{(n)}\big{(}g,z^{(n)}\big{)}=z^{(n)}$ and target map
$Z^{(n)}=\tau^{(n)}\big{(}g,z^{(n)}\big{)}=g\cdot z^{(n)}$ provided by the
prolonged action. The action of $G$ on $\mathcal{B}^{(n)}$ is given by _right-
regularization_

$\displaystyle R_{h}\big{(}g,z^{(n)}\big{)}=\big{(}g\cdot h^{-1},h\cdot
z^{(n)}\big{)}.$

Importantly, the target map $\tau^{(n)}\big{(}g,z^{(n)}\big{)}$ is _invariant_
under the right-regularized action. Therefore, the pull-back
$\big{(}\tau^{(n)}\big{)}^{*}\eta$ of any differential form $\eta$ on
$\mathcal{S}^{(n)}$ is invariant on $\mathcal{B}^{(n)}$. Since the cotangent
space $T^{*}\mathcal{B}^{(n)}=T^{*}G\times T^{*}\mathcal{S}^{(n)}$ is a direct
sum, and $G$ acts separately on its components, we may “project” any invariant
1-form on $\mathcal{B}^{(n)}$ to an invariant 1-form on $\mathcal{S}^{(n)}$.
Similarly, for higher order forms, we have the direct sums

$\displaystyle\bigwedge\nolimits^{k}T^{*}\mathcal{B}^{(n)}=\bigoplus_{i+j=k}\left(\bigwedge\nolimits^{i}T^{*}G\times\bigwedge\nolimits^{j}T^{*}\mathcal{S}^{(n)}\right)\\!,$

which the right-regularized action preserves, and so we also have an invariant
projection

$\displaystyle\pi_{J}\colon\
\bigwedge\nolimits^{k}T^{*}\mathcal{B}^{(n)}\to\bigwedge\nolimits^{k}T^{*}\mathcal{S}^{(n)}$

that maps invariant $k$-forms on $\mathcal{B}^{(n)}$ to invariant $k$-forms on
$\mathcal{S}^{(n)}$. In practice we apply $\pi_{J}$ by writing a $k$-form
$\eta$ on $\mathcal{B}^{(n)}$ as a direct sum of wedge products of forms on
$G$ and $\mathcal{S}^{(n)}$ and then set all $T^{*}G$-terms (which in our case
will be the Maurer–Cartan forms) to zero.

Given a differential form $\eta$ on $\mathcal{S}^{(n)}$, we introduce the
_lift map_

$\displaystyle\bm{\lambda}(\eta):=\pi_{J}\big{(}\tau^{(n)}\big{)}^{*}\eta,$
(2.2)

which returns an invariant form on $\mathcal{B}^{(n)}$ with only
$T^{*}\mathcal{S}^{(n)}$-components. The simplest example is given by the _$n$
th order lifted invariants_

$\displaystyle\bm{\lambda}\big{(}z^{(n)}\big{)}=g\cdot z^{(n)}=Z^{(n)}.$

######  Definition 2.1.

A _partial right moving frame_ of order $n$ is a right-invariant local
subbundle $\widehat{\rho}^{(n)}\subset\mathcal{B}^{(n)}$, meaning that
$R_{h}\big{(}\widehat{\rho}^{(n)}\big{)}\subset\widehat{\rho}^{(n)}$ for all
$h\in G$.

In practice, a partial moving frame is obtained by choosing a cross-section
$\mathcal{K}^{(n)}\subset\mathcal{S}^{(n)}$ transversed to the prolonged group
action. Then
$\widehat{\rho}^{(n)}=\big{(}\tau^{(n)}\big{)}^{-1}(\mathcal{K}^{(n)})$ is a
partial moving frame of order $n$.

######  Remark 2.2.

We note that as opposed to the standard moving frame definition [8] a partial
moving frame allows for some of the group parameters to not be normalized.
More precisely, if $\mathcal{K}^{(n)}\subset\mathcal{S}^{(n)}$ has codimension
$k_{n}$, then $\widehat{\rho}^{(n)}$ also has codimension $k_{n}$, which
implies that $r-k_{n}$ group parameters remain unnormalized.

Given a partial moving frame $\widehat{\rho}^{(n)}$, we introduce the
_partially normalized invariants_

$\displaystyle\widehat{Z}^{(n)}=\big{(}\widehat{\rho}^{(n)}\big{)}^{*}\big{[}\bm{\lambda}\big{(}z^{(n)}\big{)}\big{]}.$

The partially normalized invariants are obtained by substituting the
normalized group parameters into the lifted invariants $Z^{(n)}$. To simplify
the notation in Sections 3, 4, and 5, we do not include the hat notation over
the partially normalized invariants. We hope that the context will make it
clear that we are working with the partially normalized invariants.

###  2.2 Recurrence relations

The recurrence relations introduced in this section is one of the most
important contributions of [8] to the method of moving frames. These equations
unlock the structure of the algebra of differential invariants (and more
generally that of differential forms). One of the key aspects of these
equations is that they can be derived without the coordinate expressions for
the (partial) moving frame, the differential invariants, and the invariant
differential forms.

First, a coframe on $T^{*}\mathcal{B}^{(\infty)}$ is given by a basis of
Maurer–Cartan forms $\mu^{1},\dots,\mu^{r}$, the horizontal forms ${\rm
d}x^{1},\dots,{\rm d}x^{p}$, and the basic contact one-forms
$\theta^{\alpha}_{J}={\rm d}u^{\alpha}_{J}-u^{\alpha}_{J,j}{\rm d}x^{j}$.
Throughout this paper we use the Einstein summation convention, where
summation occurs over repeated indices. Since all our computations are
performed modulo contact forms, these are omitted from this point forward.

Applying the lift map (2.2) to the horizontal coframe results in the invariant
one-forms

$\displaystyle\omega^{i}=\bm{\lambda}\big{(}{\rm d}x^{i}\big{)}$

called _lifted horizontal forms_.

Next, let

$\displaystyle\mathbf{v}_{\nu}=\xi^{i}_{\nu}(z)\frac{\partial}{\partial
x^{i}}+\phi^{\alpha}_{\nu}(z)\frac{\partial}{\partial
u^{\alpha}},\qquad\nu=1,\dots,r=\dim G,$

be a basis of infinitesimal generators dual to the Maurer–Cartan form
$\mu^{1},\dots,\mu^{r}$. Then the recurrence relations for the lifted
invariants measure the extend to which
$d\circ\bm{\lambda}\neq\bm{\lambda}\circ d$. These equations are

$\displaystyle{\rm d}X^{i}=\omega^{i}+\xi^{i}_{\nu}(Z)\mu^{\nu},$
$\displaystyle{\rm
d}U^{\alpha}_{J}=U_{J,j}^{\alpha}\omega^{j}+\phi^{\alpha;J}_{\nu}\big{(}Z^{(\\#J)}\big{)}\mu^{\nu},$
(2.3)

where the prolonged vector field coefficients are given by the standard
recursive formula

$\displaystyle\phi^{\alpha;J,j}_{\nu}=D_{j}\phi^{\alpha;J}_{\nu}-\big{(}D_{j}\xi^{i}_{\nu}\big{)}\cdot
u^{\alpha}_{J,i}.$

Given a partial moving frame $\widehat{\rho}^{(n)}$, which we can consider to
be in $\mathcal{B}^{(\infty)}$ using the natural inclusion
$i^{(n)}\colon\mathcal{B}^{(n)}\hookrightarrow\mathcal{B}^{(\infty)}$, we can
then pull-back the lifted recurrence relations (2.3) by $\widehat{\rho}^{(n)}$
to obtain the recurrence relations for the partially normalized invariants

$\displaystyle{\rm
d}\widehat{X}^{i}=\widehat{\omega}^{i}+\xi^{i}_{\nu}\big{(}\widehat{Z}\big{)}\widehat{\mu}^{\nu},$
$\displaystyle{\rm
d}\widehat{U}^{\alpha}_{J}=\widehat{U}_{J,j}^{\alpha}\widehat{\omega}^{j}+\phi^{\alpha;J}_{\nu}\big{(}\widehat{Z}^{(\\#J)}\big{)}\widehat{\mu}^{\nu},$

where

$\displaystyle\widehat{\omega}^{i}=\big{(}\widehat{\rho}^{(n)}\big{)}^{*}\omega^{i}\qquad\text{and}\qquad\widehat{\mu}^{\nu}=\big{(}\widehat{\rho}^{(n)}\big{)}^{*}\mu^{\nu}$

are the partially normalized horizontal one-forms and the partially normalized
Maurer–Cartan forms, respectively.

######  Remark 2.3.

As in the standard moving frame implementation, the symbolic expressions for
the partially normalized Maurer–Cartan forms can be deduced from the
recurrence relations for the phantom invariants, i.e., the lifted invariants
that are equal to constant values by virtue of the moving frame construction.
We refer the reader to [8] for more detail.

######  Remark 2.4.

If the prolonged action becomes free on $\mathcal{S}^{(n)}$, for a
sufficiently large $n$, we note that the partial moving frame construction
outlined above reproduces the usual moving frame construction first introduced
in [8]. We note that depending on $\mathcal{S}^{(n)}$, freeness cannot always
be achieved and this even if the action is locally effective on subsets. Thus,
Proposition 9.6 of [8] holds on regular subsets of the submanifold jet space
but not necessarily on invariant subbundles of the form (2.1). When freeness
cannot be attained, the most one can construct is a partial moving frame.

###  2.3 Recursive moving frames

For a detailed exposition of the recursive moving frame implementation, we
refer the reader to the original work [24]. One of the main issues of the
standard moving frame implementation is that it first requires computing the
prolonged action, which relies on implicit differentiation, and can lead to
unwieldy expressions that limit the method’s practical scope and
implementation. This holds true even when using symbolic softwares such as
Mathematica, Maple, or Sage. Some of the results obtained in this paper are a
prime example of this fact. Indeed, we implemented the standard moving frame
machinery in Mathematica and in some cases the software was unable to solve
the normalization equations that produces the moving frame. In those cases we
had to revert to the recursive implementation.

The idea of the recursive moving frame method is, in the spirit of Cartan’s
original approach, to recursively normalize group parameters at a given order
before prolonging the action to the next higher order jet space. Instead of
using implicit differentiation to compute the prolonged action, the key idea
of the recursive moving frame implementation is to use the recurrence formulas
and the expressions for the Maurer–Cartan forms

$\displaystyle\bm{\mu}={\rm d}g\cdot g^{-1}.$ (2.4)

To illustrate the recursive moving frame method, assume the prolonged action
up to order $n$ is known and that a partial moving frame
$\widehat{\rho}^{(n)}$ has been computed using a cross-section
$\mathcal{K}^{(n)}\subset\mathcal{S}^{(n)}$. Assuming, for simplicity, that
$\mathcal{K}^{(n)}$ is a coordinate cross-section, suppose $u^{\alpha}_{J}=c$,
with $\\#J=n$ is one of the defining equation of $\mathcal{K}^{(n)}$. Then
$\widehat{U}^{\alpha}_{J}=c$ is a phantom invariant and its recurrence
relation yields

$\displaystyle 0={\rm
d}c=\widehat{U}^{\alpha}_{J,j}\widehat{\omega}^{j}+\phi^{\alpha;J}_{\nu}\big{(}\widehat{Z}^{(n)}\big{)}\widehat{\mu}^{\nu}$

so that

$\displaystyle\widehat{U}^{\alpha}_{J,j}\widehat{\omega}^{j}=-\phi^{\alpha;J}_{\nu}\big{(}\widehat{Z}^{(n)}\big{)}\widehat{\mu}^{\nu}.$
(2.5)

By assumption, coordinate expressions for
$\phi^{\alpha;J}_{\nu}(\widehat{Z}^{(n)})$ are known, since the prolonged
action up to order $n$ has been computed, and the partially normalized
Maurer–Cartan forms $\widehat{\mu}^{\nu}$ can be found by substituting the
group normalizations into (2.4). Expressing the right-hand side of (2.5) as a
linear combination of the partially normalized horizontal forms
$\widehat{\omega}^{i}$, we are able to obtain expressions for the order $n+1$
partially normalized invariants $\widehat{U}^{\alpha}_{J,j}$.

###  2.4 The algebra of differential invariants

Assume a moving frame is known or that a partial moving frame has been
computed with no possibility of further group parameter normalizations. Dual
to the invariant horizontal forms $\omega^{i}$ are the _invariant total
derivative operators_

$\displaystyle\mathcal{D}_{i}=\widehat{W}^{j}_{i}D_{i},\qquad\text{where}\quad\big{(}\widehat{W}^{j}_{j}\big{)}=\big{(}\widehat{\rho}^{(n)}\big{)}^{*}\big{(}D_{j}X^{i}\big{)}^{-1}.$
(2.6)

Now, let

$\displaystyle{\rm
d}\omega^{i}=C^{i}_{jk}\omega^{j}\wedge\omega^{k}\mod(\text{unnormalized
Maurer--Cartan forms})$ (2.7)

be the structure equations among the invariant horizontal forms. These
equations can be obtained symbolically by extending the recurrence relations
(2.3) to differential forms as done in [13]. Given (2.7), the commutation
relations among the invariant total derivative operators are

$\displaystyle[\mathcal{D}_{j},\mathcal{D}_{k}]=-C^{i}_{jk}\mathcal{D}_{i}.$
(2.8)

Fix $j$, $k$ in (2.8) and apply the commutation relation to $p$ invariants
$I_{1},\dots,I_{p}$ to obtain
$[\mathcal{D}_{j},\mathcal{D}_{k}]I_{\ell}=-C^{i}_{jk}\mathcal{D}_{i}I_{\ell}$.
In matrix form

$\displaystyle[\mathcal{D}_{j},\mathcal{D}_{k}]I=-\mathcal{D}IC_{jk},$

where
$[\mathcal{D}_{j},\mathcal{D}_{k}]I=([\mathcal{D}_{j},\mathcal{D}_{k}]I_{1},\dots,[\mathcal{D}_{j},\mathcal{D}_{k}]I_{p})^{\rm
T}$, $\mathcal{D}I=(\mathcal{D}_{i}I_{\ell})$, and
$C_{jk}=\big{(}C^{1}_{jk},\dots,C^{p}_{jk}\big{)}^{\rm T}$. If
$\det\mathcal{D}I\not\equiv 0$, then one can solve for $C_{ij}$

$\displaystyle C_{jk}=-(\mathcal{D}I)^{-1}[\mathcal{D}_{j},\mathcal{D}_{k}]I,$
(2.9)

which allows one to express the commutator invariants $C_{jk}$ in terms of
$I=(I_{1},\dots,I_{p})$ and its invariant derivatives. This is what we refer
to as the _commutator trick_. Notice that given a single invariant $I_{1}$, we
could have set $I_{i}:=\mathcal{D}_{k_{i}}^{\ell_{i}}I_{1}$, with $1\leq
k_{i}\leq p$ and $\ell_{i}\geq 0$, in order to write the commutator invariants
$C_{jk}$ as functions of a single invariant and its invariant derivatives.
This observation plays a key role in showing that the algebras of differential
invariants for Euclidean, equi-affine, conformal, and projective surfaces are
generically generated by a single invariant [11, 23, 25]. The commutator trick
will also be used in this paper to show that certain algebras of differential
invariants are generated by a single invariant.

We now recall important results about the algebra of differential invariants
that can be found in [8, 22].

######  Proposition 2.5.

The normalized invariants $\widehat{Z}^{(n)}$ provide a complete set of
differential invariants of order $\leq n$.

By the _replacement principle_ [8, 17], if $I\big{(}z^{(n)}\big{)}$ is a
differential invariant, then it can be written in terms of the normalized
invariants as $I=I\big{(}\widehat{Z}^{(n)}\big{)}$, which is obtained by
replacing the jet coordinates $z^{(n)}$ by their corresponding normalized
invariants $\widehat{Z}^{(n)}$.

######  Definition 2.6.

A set of invariants $\bm{I}_{{\rm gen}}=\\{I_{1},\dots,I_{\ell}\\}$ is said to
_generate_ the algebra of differential invariants if any differential
invariant can be expressed in terms of $\bm{I}_{{\rm gen}}$ and its invariant
derivatives (2.6) of any order.

From Proposition 2.5 it follows that if one can show that the normalized
invariants $\widehat{Z}^{(\infty)}$ can be written in terms of a set of
invariants $\bm{I}_{{\rm gen}}$ and its invariant derivatives, then
$\bm{I}_{{\rm gen}}$ is a generating set for the algebra of differential
invariants.

######  Theorem 2.7.

Given a moving frame $\widehat{\rho}^{(n)}$, the normalized invariants
$\bm{I}_{{\rm gen}}=\big{\\{}\widehat{Z}^{(n+1)}\big{\\}}$ form a generating
set of differential invariants.

The generating set in Theorem 2.7 is not necessarily minimal. By that we mean
that it might be possible to remove certain non-phantom invariants and still
obtain a generating set. To this day, there is no known result that stipulates
how small the generating set can be. But if one can show that the invariants
$\bm{I}_{{\rm gen}}=\big{\\{}\widehat{Z}^{(n+1)}\big{\\}}$ can be expressed in
terms of a single invariant $I$ and its invariant derivatives
$\mathcal{D}_{1},\dots,\mathcal{D}_{p}$, then the algebra of differential
invariants is generated by a single function. This is the approach used in the
following sections to show that the various differential invariant algebras
are generated by a single invariant.

##  3 Affine action and low-order normalizations

In the following, we consider surfaces $S\subset\mathbb{R}^{3}$, which we
assume are locally given a graphs of functions:

$\displaystyle S=\\{z=(x,y,u(x,y))\\}\subset\mathbb{R}^{3}.$

We are interested in the action of the affine group ${\rm
A}(3,\mathbb{R})={\rm GL}(3,\mathbb{R})\ltimes\mathbb{R}^{3}$ on these
surfaces given by

$\displaystyle Z=Az+b,\qquad\text{where}\quad A\in{\rm
GL}(3,\mathbb{R})\quad\text{and}\quad b\in\mathbb{R}^{3}.$

A basis for the algebra of infinitesimal generators is provided by

$\displaystyle\mathbf{v}_{xx}=x\frac{\partial}{\partial x},\quad\
\mathbf{v}_{xy}=y\frac{\partial}{\partial x},\quad\
\mathbf{v}_{xu}=u\frac{\partial}{\partial x},\quad\
\mathbf{v}_{yx}=x\frac{\partial}{\partial y},\quad\
\mathbf{v}_{yy}=y\frac{\partial}{\partial y},\quad\
\mathbf{v}_{yu}=u\frac{\partial}{\partial y},$
$\displaystyle\mathbf{v}_{ux}=x\frac{\partial}{\partial u},\quad\
\mathbf{v}_{uy}=y\frac{\partial}{\partial u},\quad\
\mathbf{v}_{uu}=u\frac{\partial}{\partial u},\quad\
\mathbf{v}_{x}=\frac{\partial}{\partial x},\quad\
\mathbf{v}_{y}=\frac{\partial}{\partial y},\quad\
\mathbf{v}_{u}=\frac{\partial}{\partial u}.$

Let

$\displaystyle\bm{\mu}=\begin{bmatrix}\mu&\nu\\\
0&0\end{bmatrix}\qquad\text{with}\quad\mu=\begin{bmatrix}\mu^{xx}&\mu^{xy}&\mu^{xu}\\\
\mu^{yx}&\mu^{yy}&\mu^{yu}\\\
\mu^{ux}&\mu^{uy}&\mu^{uu}\end{bmatrix}\quad\text{and}\quad\nu=\begin{bmatrix}\mu^{x}\\\
\mu^{y}\\\ \mu^{u}\end{bmatrix}$

denote a basis of Maurer–Cartan forms with structure equations

$\displaystyle{\rm d}\mu=-\mu\wedge\mu,\qquad{\rm d}\nu=-\mu\wedge\nu.$

Then the order zero recurrence relations for the lifted invariants are

$\displaystyle{\rm d}X=\omega^{x}+X\mu^{xx}+Y\mu^{xy}+U\mu^{xu}+\mu^{x},$
$\displaystyle{\rm d}Y=\omega^{y}+X\mu^{yx}+Y\mu^{yy}+U\mu^{yu}+\mu^{y},$
$\displaystyle{\rm d}U=U_{j}\omega^{j}+X\mu^{ux}+Y\mu^{uy}+U\mu^{uu}+\mu^{u},$

while for $k+\ell\geq 1$,

$\displaystyle{\rm
d}U_{X^{k}Y^{\ell}}=U_{X^{k}Y^{\ell}j}\omega^{j}-kU_{X^{k}Y^{\ell}}\mu^{xx}-\ell
U_{X^{k+1}Y^{\ell-1}}\mu^{xy}-kU_{X^{k-1}Y^{\ell+1}}\mu^{yx}-\ell
U_{X^{k}Y^{\ell}}\mu^{yy}$ $\displaystyle\hphantom{{\rm
d}U_{X^{k}Y^{\ell}}=}{}+U_{X^{k}Y^{\ell}}\mu^{uu}+\delta_{1k}\delta_{0\ell}\mu^{ux}+\delta_{0k}\delta_{1\ell}\mu^{uy}$
$\displaystyle\hphantom{{\rm
d}U_{X^{k}Y^{\ell}}=}{}-\sum_{\begin{subarray}{c}0\leq i\leq k\\\ 0\leq
j\leq\ell\\\
(i,j)\neq(k,\ell)\end{subarray}}\binom{k}{i}\binom{\ell}{j}\big{[}U_{X^{k-i}Y^{\ell-j}}U_{X^{i+1}Y^{j}}\mu^{xu}+U_{X^{k-i}Y^{\ell-j}}U_{X^{i}Y^{j+1}}\mu^{yu}\big{]},$

where there is no summation over $k$ and $\ell$, and $\delta_{ij}$ denotes the
Kronecker delta function.

Since the action is transitive on ${\rm J}^{(1)}$, we can set

$\displaystyle X=Y=U=U_{X}=U_{Y}=0.$ (3.1)

In other words, we can choose the cross-section
$\mathcal{K}^{(1)}=\\{x=y=u=u_{x}=u_{y}=0\\}\subset{\rm J}^{(1)}$. The
recurrence relations for these phantom invariants are

$\displaystyle 0=\omega^{x}+\mu^{x},\qquad 0=\omega^{y}+\mu^{y},\qquad
0=\mu^{u},\qquad 0=U_{Xj}\omega^{j}+\mu^{ux},\qquad
0=U_{Yj}\omega^{j}+\mu^{uy}.$

As mentioned in Section 2.1, from this point onward we omit the use of the hat
notation to denote partially normalized quantities. Solving for the
Maurer–Cartan forms yields

$\displaystyle\mu^{x}=-\omega^{x},\qquad\mu^{y}=-\omega^{y},\qquad\mu^{u}=0,\qquad\mu^{ux}=-U_{Xj}\omega^{j},\qquad\mu^{uy}=-U_{Yj}\omega^{j}.$
(3.2)

Taking into account the order 0 and 1 normalizations (3.1), and the normalized
Maurer–Cartan forms (3.2), the recurrence relations for the order 2 partially
normalized invariants are

$\displaystyle{\rm
d}U_{XX}=U_{XXj}\omega^{j}+U_{XX}\big{(}\mu^{uu}-2\mu^{xx}\big{)}-2U_{XY}\mu^{yx},$
$\displaystyle{\rm
d}U_{XY}=U_{XYj}\omega^{j}-U_{XX}\mu^{xy}+U_{XY}\big{(}\mu^{uu}-\mu^{xx}-\mu^{yy}\big{)}-U_{YY}\mu^{yx},$
$\displaystyle{\rm
d}U_{YY}=U_{YYj}\omega^{j}+U_{YY}(\mu^{uu}-2\mu^{yy})-2U_{XY}\mu^{xy}.$ (3.3)

Consider the partially normalized lifted Hessian determinant

$\displaystyle H=U_{XX}U_{YY}-U_{XY}^{2}.$

Since

$\displaystyle{\rm
d}H=2H\big{(}\mu^{uu}-\mu^{xx}-\mu^{yy}\big{)}\mod(\omega^{x},\omega^{y}),$

we conclude that $H$ is a relative invariant. To obtain an expression for $H$,
we introduce the determinant

$\displaystyle|D\mathbf{X}|=\det\begin{bmatrix}X_{x}&X_{y}\\\
Y_{x}&Y_{y}\end{bmatrix}=\det\begin{bmatrix}a_{11}+a_{13}u_{x}&a_{12}+a_{13}u_{y}\\\
a_{21}+a_{23}u_{y}&a_{22}+a_{23}u_{y}\end{bmatrix}$ (3.4)

and the Hessian determinant $h=u_{xx}u_{yy}-u_{xy}^{2}$. Then

$\displaystyle H=\frac{a_{33}^{2}}{|D\mathbf{X}|^{2}}h.$

######  Definition 3.1.

A point $\big{(}x,y,u^{(2)}\big{)}$ of $S^{(2)}\in{\rm J}^{(2)}$ is said to be

  * •

_elliptic_ if $h>0$,

  * •

_hyperbolic_ if $h<0$,

  * •

_parabolic_ if $h=0$.

The remaining analysis depends on the sign of the Hessian determinant. Since
most results for elliptic and hyperbolic points are similar, these two cases
are combined together in the next section. The case of parabolic points is
considered in Section 5.

##  4 Elliptic and hyperbolic points

In this section we work under the assumption that

$\displaystyle H=\epsilon=\pm 1,$

with $\epsilon=1$ corresponding to the elliptic case and $\epsilon=-1$ to
hyperbolic points. From the recurrence relations (3.3), we conclude that it is
possible to set

$\displaystyle U_{XX}=1,\qquad U_{YY}=\epsilon,\qquad U_{XY}=0.$ (4.1)

######  Remark 4.1.

In Cartesian coordinates, the normalization equations (4.1) are quadratic in
the group parameters. Therefore, in the process of constructing a moving frame
there is a choice of sign that needs to be made. But since (4.1) holds, no
matter the choice made, this does not affect the algebra of differential
invariants of the surface and as such is not important for our purpose. Thus,
as it is customary [25], in the following we omit such ambiguity.

After the normalizations (4.1) have been performed, the recurrence relations
for the order 3 partially normalized invariants are

$\displaystyle{\rm
d}U_{X^{3}}=-3\mu^{xu}-\frac{U_{X^{3}}}{2}\mu^{uu}\mod(\omega^{x},\omega^{y}),$
$\displaystyle{\rm d}U_{X^{2}Y}=-\epsilon\mu^{yu}+\epsilon
U_{X^{3}}\mu^{yx}-2U_{XY^{2}}\mu^{yx}-\frac{U_{X^{2}Y}}{2}\mu^{uu}\mod(\omega^{x},\omega^{y}),$
$\displaystyle{\rm d}U_{XY^{2}}=-\epsilon\mu^{xu}+2\epsilon
U_{X^{2}Y}\mu^{yx}-U_{Y^{3}}\mu^{yx}-\frac{U_{XY^{2}}}{2}\mu^{uu}\mod(\omega^{x},\omega^{y}),$
$\displaystyle{\rm d}U_{Y^{3}}=-3\mu^{yu}+3\epsilon
U_{XY^{2}}\mu^{yx}-\frac{U_{Y^{3}}}{2}\mu^{uu}\mod(\omega^{x},\omega^{y}).$

Consistent with normalizations performed for elliptic and hyperbolic surfaces
in equi-affine geometry [23], we set

$\displaystyle U_{X^{3}}+\epsilon U_{XY^{2}}=U_{Y^{3}}+\epsilon U_{X^{2}Y}=0$

and solve for $U_{XY^{2}}$ and $U_{X^{2}Y}$. We are then left with $U_{X^{3}}$
and $U_{Y^{3}}$, whose recurrence relations are

$\displaystyle{\rm d}U_{X^{3}}=3\epsilon
U_{Y^{3}}\mu^{yx}-\frac{U_{X^{3}}}{2}\mu^{uu}\mod(\omega^{x},\omega^{y}),$
$\displaystyle{\rm
d}U_{Y^{3}}=-3U_{X^{3}}\mu^{yx}-\frac{U_{Y^{3}}}{2}\mu^{uu}\mod(\omega^{x},\omega^{y}).$

The extent to which one can solve for the partially normalized Maurer–Cartan
forms $\mu^{yx}$ and $\mu^{uu}$ depends on the determinant

$\displaystyle\det\begin{bmatrix}3\epsilon
U_{Y^{3}}&-\dfrac{U_{X^{3}}}{2}\\\\[6.45831pt]
-3U_{X^{3}}&-\dfrac{U_{Y^{3}}}{2}\end{bmatrix}=-\frac{3}{2}\big{(}U_{X^{3}}^{2}+\epsilon
U_{Y^{3}}^{2}\big{)}=-\frac{3}{2}P_{\epsilon}.$

We note that $P_{\epsilon}$ is a relative invariant as

$\displaystyle{\rm d}P_{\epsilon}=-P_{\epsilon}\mu^{uu}.$

In fact, $P_{\epsilon}=\frac{P}{a_{33}}$, where $P$ is the equi-affine Pick
invariant

$\displaystyle
P=\frac{1}{16\big{(}u_{xx}u_{yy}-u_{xy}^{2}\big{)}^{3}}\big{[}6u_{xx}u_{xy}u_{yy}u_{xxx}u_{yyy}-6_{xx}u_{yy}^{2}u_{xxx}u_{xyy}-18u_{xx}u_{xy}u_{yy}u_{xxy}u_{xyy}$
$\displaystyle\hphantom{P=}{}+12u_{xx}u_{xy}^{2}u_{xxy}u_{yyy}-6u_{xx}^{2}u_{yy}u_{xxy}u_{yyy}+9u_{xx}u_{yy}^{2}u_{xxy}^{2}-6u_{xx}^{2}u_{xy}u_{xyy}u_{yyy}$
$\displaystyle\hphantom{P=}{}+9u_{xx}^{2}u_{yy}u_{xyy}^{2}+u_{xx}^{3}u_{yyy}^{2}-6u_{xy}u_{yy}^{2}u_{xxx}u_{xxy}+12u_{xy}^{2}u_{yy}u_{xxx}u_{xyy}$
$\displaystyle\hphantom{P=}{}-8u_{xy}^{3}u_{xxx}u_{yyy}+u_{yy}^{3}u_{xxx}^{2}\big{]}.$

We now need to distinguish the cases where $P_{\epsilon}\equiv 0$ is
identically zero and where $P_{\epsilon}\neq 0$ does not vanish. In the
elliptic case, we note that if $P_{1}\equiv 0$, then $U_{X^{3}}\equiv
U_{Y^{3}}\equiv 0$. On the other hand, in the hyperbolic case, when
$P_{-1}\equiv 0$, we have that $U_{Y^{3}}\equiv\pm U_{X^{3}}$. But, we observe
that under the change of variables $(x,y,u)\mapsto(x,-y,u)$, we can always
assume that $U_{Y^{3}}=-U_{X^{3}}$. Therefore, at hyperbolic points there are
two cases to consider, either $U_{X^{3}}\equiv 0$ or $U_{X^{3}}\neq 0$. We
combine the different cases as follows:

$\displaystyle{\bf EH.1}\colon\ P_{\epsilon}\neq 0,\qquad{\bf EH.2}\colon\
U_{X^{3}}\equiv U_{Y^{3}}\equiv 0,\qquad{\bf H.3}\colon\ U_{Y^{3}}\equiv-
U_{X^{3}}\neq 0.$

We note that cases EH.1 and EH.2 hold for both elliptic and hyperbolic points
whereas case H.3 is only for hyperbolic points. In local coordinates, since

$\displaystyle U_{XXX}=\frac{C_{1}\big{(}3\epsilon
a_{33}u_{xx}Y_{x}-4Y_{x}^{3}\big{)}-C_{2}\big{(}\epsilon
a_{33}u_{xx}-4Y_{x}^{2}\big{)}\sqrt{|h|}\sqrt{a_{33}u_{xx}-\epsilon
Y_{x}^{2}}}{4a_{33}^{2}u_{xx}^{3}|h|^{3/2}},$ $\displaystyle
U_{YYY}=\frac{C_{1}\big{(}a_{33}u_{xx}-4\epsilon
Y_{x}^{2}\big{)}\sqrt{a_{33}u_{xx}-\epsilon
Y_{x}^{2}}+C_{2}\sqrt{|h|}\big{(}3\epsilon
a_{33}u_{xx}Y_{x}-4Y_{x}^{3}\big{)}}{4a_{33}^{2}u_{xx}^{3}|h|^{3/2}},$

and

$\displaystyle
P_{\epsilon}=\frac{C_{1}^{2}+hC_{2}^{2}}{16a_{33}u_{xx}^{3}h^{3}},$

where

$\displaystyle
C_{1}=6u_{xx}u_{xy}^{2}u_{xxy}-4u_{xy}^{3}u_{xxx}-3u_{xx}^{2}u_{xy}u_{xyy}-3u_{xx}^{2}u_{yy}u_{xxy}+3u_{xx}u_{xy}u_{yy}u_{xxx}+u_{xx}^{3}u_{yyy},$
$\displaystyle
C_{2}=-6u_{xx}u_{xy}u_{xxy}+4u_{xy}^{2}u_{xxx}+3u_{xx}^{2}u_{xyy}-u_{xx}u_{yy}u_{xxx}.$

the three cases can be restated as

$\displaystyle{\bf EH.1}\colon\ C_{1}^{2}+hC_{2}^{2}\neq 0,\qquad{\bf
EH.2}\colon\ C_{1}\equiv C_{2}\equiv 0,\qquad{\bf H.3}\colon\ C_{1}\equiv-
C_{2}\sqrt{|h|}\neq 0.$

######  Remark 4.2.

We remark that the expressions for $U_{XXX}$ and $U_{YYY}$ hold provided
$u_{xx}\neq 0$. From this point forward, we always work on the open dense
subset of the jet space where $u_{xx}\neq 0$.

###  4.1 Case EH.1

When $P_{\epsilon}\neq 0$, it is possible to set

$\displaystyle U_{X^{3}}=1,\qquad U_{Y^{3}}=0.$

According to Theorem 2.7, the order 4 differential invariants

$\displaystyle U_{X^{4}},\qquad U_{X^{3}Y},\qquad U_{X^{2}Y^{2}},\qquad
U_{XY^{3}},\qquad U_{Y^{4}},$

form a complete set of generating invariants. We now show in fact that the
algebra of differential invariants is generically generated by the single
invariant $I_{1}=U_{Y^{4}}$. First, the structure equations for the invariant
coframe $\omega^{x}$, $\omega^{y}$ are

$\displaystyle{\rm
d}\omega^{x}=\frac{2\epsilon}{3}U_{XY^{3}}\omega^{x}\wedge\omega^{y},\qquad{\rm
d}\omega^{y}=\frac{1}{12}\big{(}3U_{X^{4}}-6\epsilon
U_{X^{2}Y^{2}}-U_{Y^{4}}\big{)}\omega^{x}\wedge\omega^{y}.$

Therefore, the Lie bracket of the invariant total derivative operators is

$\displaystyle[\mathcal{D}_{x},\mathcal{D}_{y}]=-\frac{2\epsilon}{3}U_{XY^{3}}\mathcal{D}_{x}-\frac{1}{12}\big{(}3U_{X^{4}}-6\epsilon
U_{X^{2}Y^{2}}-U_{Y^{4}}\big{)}\mathcal{D}_{y}.$

Using the commutator trick (2.9), we can generically solve for
$I_{2}=U_{XY^{3}}$ and $I_{3}=U_{X^{4}}-2\epsilon U_{X^{2}Y^{2}}$ in terms of
$I_{1}$ and its invariant derivatives. Indeed, applying the commutator trick
to $I_{1}$ and $\mathcal{D}I_{1}$, where $\mathcal{D}$ is a nontrivial
invariant total derivative operator, we find that

$\displaystyle\begin{pmatrix}-\frac{2\epsilon I_{2}}{3}\\\\[2.15277pt]
\frac{I_{1}}{12}-\frac{I_{3}}{4}\end{pmatrix}=\begin{pmatrix}\mathcal{D}_{x}I_{1}&\mathcal{D}_{y}I_{1}\\\\[2.15277pt]
\mathcal{D}_{x}\mathcal{D}I_{1}&\mathcal{D}_{y}\mathcal{D}I_{1}\end{pmatrix}^{-1}\begin{pmatrix}[\mathcal{D}_{x},\mathcal{D}_{y}]I_{1}\\\\[2.15277pt]
[\mathcal{D}_{x},\mathcal{D}_{y}]\mathcal{D}I_{1}\end{pmatrix},$

which can be solved for $I_{2}$ and $I_{3}$ provided that

$\displaystyle\mathcal{D}_{x}I_{1}\cdot\mathcal{D}_{y}\mathcal{D}I_{1}-\mathcal{D}_{y}I_{1}\cdot\mathcal{D}_{x}\mathcal{D}I_{1}\neq
0.$

Next, consider the syzygy

$\displaystyle\mathcal{D}_{x}I_{1}-\mathcal{D}_{y}I_{2}=\frac{3}{2}I_{3}-\frac{7\epsilon}{6}I_{2}^{2}-\frac{1}{2}I_{1}I_{3}+\frac{1}{2}I_{1}^{2}+\frac{1}{4}\big{(}3U_{X^{2}Y^{2}}^{2}+6\epsilon
U_{X^{2}Y^{2}}-2U_{X^{3}Y}U_{XY^{3}}\big{)}.$ (4.2)

This suggests the introduction of the fourth order invariant

$\displaystyle
I_{4}=3U_{X^{2}Y^{2}}^{2}+6U_{X^{2}Y^{2}}-2U_{X^{3}Y}U_{XY^{3}}.$

Also, from (4.2) is follows that $I_{4}$ can be expressed in terms of $I_{1}$,
$I_{2}$, $I_{3}$ and their invariant derivatives. Since $I_{2}$ and $I_{3}$
can be expressed in terms of $I_{1}$ and its invariant derivatives, the same
holds true for $I_{4}$.

Now, considering the fifth order invariants $\mathcal{D}_{i}I_{j}$, we find,
using Mathematica, the syzygy

$\displaystyle-216I_{2}\mathcal{D}_{x}I_{2}-108\epsilon
I_{2}\mathcal{D}_{y}I_{3}+36\epsilon
I_{2}\mathcal{D}_{y}I_{4}+216I_{2}^{2}-36I_{2}^{2}\mathcal{D}_{x}I_{3}+12I_{2}^{2}\mathcal{D}_{x}I_{4}+54I_{1}I_{2}^{2}$
$\displaystyle\qquad{}+48\epsilon
I_{2}^{3}\mathcal{D}_{x}I_{2}+24I_{2}^{3}\mathcal{D}_{y}I_{3}+36\epsilon
I_{2}^{4}-4\epsilon
I_{1}I_{2}^{4}-108I_{2}^{2}I_{3}+6I_{1}I_{2}^{2}I_{3}-10\epsilon
I_{2}^{4}I_{3}$ $\displaystyle\qquad{}-36\epsilon
I_{4}\mathcal{D}_{y}I_{2}-9\epsilon
I_{1}I_{4}-12I_{2}I_{4}\mathcal{D}_{x}I_{2}-30I_{2}^{2}I_{4}-2I_{1}I_{2}^{2}I_{4}+3I_{2}^{2}I_{3}I_{4}$
$\displaystyle\qquad{}+\big{(}216\epsilon\mathcal{D}_{y}I_{2}+54\epsilon
I_{1}+72I_{2}\mathcal{D}_{x}I_{2}-432\epsilon
I_{2}\mathcal{D}_{x}I_{2}-108I_{2}\mathcal{D}_{y}I_{3}+36I_{2}\mathcal{D}_{y}I_{4}$
$\displaystyle\qquad\qquad{}+180I_{2}^{2}+270\epsilon
I_{2}^{2}+12I_{1}I_{2}^{2}+66\epsilon
I_{1}I_{2}^{2}-2I_{2}^{4}-18I_{2}^{2}I_{3}-198\epsilon
I_{2}^{2}I_{3}+6\epsilon I_{1}I_{2}^{2}I_{3}$
$\displaystyle\qquad\qquad{}-27I_{4}-36I_{4}\mathcal{D}_{y}I_{2}-18I_{1}I_{4}-33\epsilon
I_{2}^{2}I_{4}\big{)}U_{X^{2}Y^{2}}$
$\displaystyle\qquad{}+\big{(}162+2166\mathcal{D}_{y}I_{2}+108\epsilon\mathcal{D}_{y}I_{2}+108I_{1}+27\epsilon
I_{1}-180I_{2}\mathcal{D}_{x}I_{2}-144I_{2}^{2}+198\epsilon I_{2}^{2}$
$\displaystyle\qquad\qquad{}+18I_{1}I_{2}^{2}-99I_{2}^{2}I_{3}-54\epsilon
I_{4}-9\epsilon I_{1}I_{4}\big{)}U_{X^{2}Y^{2}}^{2}$
$\displaystyle\qquad{}+\big{(}81+324\epsilon+108\mathcal{D}_{y}I_{2}+54I_{1}+54\epsilon
I_{1}-189\epsilon I_{2}^{2}-27I_{4}\big{)}U_{X^{2}Y^{2}}^{3}$
$\displaystyle\qquad{}+(162+162\epsilon+27\epsilon
I_{1})U_{X^{2}Y^{2}}^{4}+81U_{X^{2}Y^{2}}^{5}=0.$

This is a quintic equation in $U_{X^{2}Y^{2}}$, which can locally be solved in
terms of $I_{1}$, $I_{2}$, $I_{3}$, and $I_{4}$ and their invariant
derivatives. This shows the following results.

######  Theorem 4.3.

If the equi-affine Pick invariant $P\neq 0$ does not vanish, then the algebra
of differential invariants is generically generated by the fourth order
invariant $I_{1}=U_{Y^{4}}$.

Using the method of recursive moving frames, a coordinate expression for the
generating invariant is

$\displaystyle
U_{Y^{4}}=3+\frac{3}{P\sqrt{|h|}}(L\mathcal{D}_{y}K-K\mathcal{D}_{y}L)+\frac{3}{4P}\mathcal{D}_{y}\big{(}P\mathcal{D}_{y}(\ln|h|)\big{)}+\frac{3\epsilon}{16}\big{(}\mathcal{D}_{y}(\ln|h|)\big{)}^{2}$
$\displaystyle\hphantom{U_{Y^{4}}=}{}+\frac{3\mathcal{D}_{y}(\ln|h|)}{4P\sqrt{|h|}}(J\mathcal{D}_{y}K-I\mathcal{D}_{y}L)+\frac{3\epsilon\mathcal{D}_{x}(\ln|h|)}{4P\sqrt{|h|}}(K\mathcal{D}_{y}L-L\mathcal{D}_{y}K),$

where

$\displaystyle\mathcal{D}_{x}=\frac{1}{P\sqrt{|h|}}(LD_{x}-KD_{y}),\qquad\mathcal{D}_{y}=\frac{1}{P\sqrt{|h|}}(-JD_{x}+ID_{y})$

are invariant total derivative operators and

$\displaystyle I=\sqrt{Pu_{xx}-\epsilon K^{2}},\qquad
J=\frac{u_{xy}\sqrt{Pu_{xx}-\epsilon K^{2}}-\epsilon\sqrt{|h|}K}{u_{xx}},$
$\displaystyle L=\frac{u_{xy}K+\sqrt{|h|}\sqrt{Pu_{xx}-\epsilon
K^{2}}}{u_{xx}},$ (4.3)

with $K$ a solution to the sextic equation

$\displaystyle 16\epsilon
K^{6}-24(Pu_{xx})K^{4}+9\epsilon(Pu_{xx})^{2}K^{2}-\frac{(Pu_{xx})^{3}C_{1}^{2}}{C_{1}^{2}+hC_{2}^{2}}=0.$
(4.4)

######  Remark 4.4.

Over the real numbers, the bi-cubic equation (4.4) has one real solution for
$K^{2}$. Then, as in Remark 4.1 there is an ambiguity of sign in the
definition of $K$, but this does not affect the structure of the algebra of
differential invariants. Also, on the cross-section, equation (4.4) reduces to

$\displaystyle 16\epsilon K^{6}-24K^{4}+9\epsilon K^{2}=0$ (4.5)

so that $K^{2}=0,\frac{3}{4\epsilon}$. Perturbing (4.5) near the cross-
section, the zero root becomes positive, which implies that $K$ is defined
near the cross-section. Finally, we note that on the cross-section
$Pu_{xx}-\epsilon K^{2}=1$ so that the square roots occurring in (4.3) are
well-defined in the neighborhood of the cross-section.

###  4.2 Case EH.2

We are now assuming that $U_{X^{3}}\equiv U_{Y^{3}}\equiv 0$. Their recurrence
relations imply that

$\displaystyle U_{X^{4}}\equiv 3\epsilon U_{X^{2}Y^{2}}\equiv U_{Y^{4}},\qquad
U_{X^{3}Y}\equiv U_{XY^{3}}\equiv 0.$ (4.6)

Thus, there is only one fourth order partially normalized invariant. We
continue the analysis using the invariant

$\displaystyle
U_{X^{2}Y^{2}}=\frac{18u_{xx}u_{xy}u_{xxx}u_{xxy}\\!-\\!9u_{xx}^{2}u_{xxy}^{2}\\!-\\!(4u_{xy}^{2}\\!+\\!5u_{xx}u_{yy})u_{xxx}^{2}\\!+\\!3u_{xx}(u_{xx}u_{yy}\\!-\\!u_{xy}^{2})u_{xxxx}}{9a_{33}u_{xx}^{3}(u_{xx}u_{yy}-u_{xy}^{2})}.$

Since its recurrence relation is

$\displaystyle{\rm
d}U_{X^{2}Y^{2}}=-U_{X^{2}Y^{2}}\mu^{uu}\mod(\omega^{x},\omega^{y}),$

we now have to consider the cases

$\displaystyle{\bf EH.2.1}\colon\ U_{X^{2}Y^{2}}\neq 0,\qquad{\bf
EH.2.2}\colon\ U_{X^{2}Y^{2}}\equiv 0.$

####  4.2.1 Case EH.2.1

When $U_{X^{2}Y^{2}}\neq 0$, we can normalize

$\displaystyle U_{X^{2}Y^{2}}=1.$

From (4.6) is follows that all fourth order invariants are constant

$\displaystyle U_{X^{4}}\equiv U_{Y^{4}}\equiv 3\epsilon,\qquad
U_{X^{2}Y^{2}}=1,\qquad U_{X^{3}Y}\equiv U_{XY^{3}}\equiv 0.$

Considering their recurrence relations

$\displaystyle 0\equiv{\rm d}U_{X^{4}}=\big{(}U_{X^{5}}-3\epsilon
U_{X^{3}Y^{2}}\big{)}\omega^{x}+\big{(}U_{X^{4}Y}-3\epsilon
U_{X^{2}Y^{3}}\big{)}\omega^{y},$ $\displaystyle 0\equiv{\rm
d}U_{X^{3}Y}=U_{X^{4}Y}\omega^{x}+U_{X^{3}Y^{2}}\omega^{y},$ $\displaystyle
0\equiv{\rm d}U_{XY^{3}}=U_{X^{2}Y^{3}}\omega^{x}+U_{XY^{4}}\omega^{y},$
$\displaystyle 0\equiv{\rm d}U_{Y^{4}}=(U_{XY^{4}}-3\epsilon
U_{X^{3}Y^{2}})\omega^{x}+\big{(}U_{Y^{5}}-3\epsilon
U_{X^{2}Y^{3}}\big{)}\omega^{y},$

we find that all fifth order invariants vanish. Similarly, the recurrence
relations for the fifth order invariants imply that the sixth order invariants
are constant, and so on. Therefore, all the invariants are constant and there
are no further normalizations possible. In particular, the Maurer–Cartan form
$\mu^{yx}$ cannot be normalized. The structure equations for the coframe
$\\{\omega^{x},\omega^{y},\mu^{yx}\\}$ are

$\displaystyle{\rm d}\omega^{x}=-\epsilon\mu^{yx}\wedge\omega^{y},\qquad{\rm
d}\omega^{y}=\mu^{yx}\wedge\omega^{x},\qquad{\rm
d}\mu=\epsilon\omega^{y}\wedge\omega^{x}.$

####  4.2.2 Case EH.2.2

When $U_{X^{2}Y^{2}}\equiv 0$, the same argument as in Case EH.2.1 implies
that all higher order partially normalized invariants vanish. In this case
$\mu^{yx}$ and $\mu^{uu}$ cannot be normalized and the structure equations of
the coframe $\\{\omega^{x},\omega^{y},\mu^{yx},\mu^{uu}\\}$ are

$\displaystyle{\rm
d}\omega^{x}=\frac{1}{2}\mu^{uu}\wedge\omega^{x}-\epsilon\mu^{yx}\wedge\omega^{y},\quad\
{\rm
d}\omega^{y}=\mu^{yx}\wedge\omega^{x}+\frac{1}{2}\mu^{uu}\wedge\omega^{y},\quad\
{\rm d}\mu^{yx}=0,\quad\ {\rm d}\mu^{uu}=0.$

###  4.3 Case H.3

In this section we assume that we are at a hyperbolic point where
$\epsilon=-1$. Also, we are working under the consideration that
$U_{Y^{3}}\equiv-U_{X^{3}}\neq 0$. Thus, it is possible to normalize
$U_{X^{3}}=1$. At order 4, the recurrence relation for
$U_{Y^{3}}+U_{X^{3}}\equiv 0$, yields the equalities

$\displaystyle U_{XY^{3}}\equiv-U_{X^{4}}-3U_{X^{2}Y^{2}}-3U_{X^{3}Y},\qquad
U_{Y^{4}}\equiv 3U_{X^{4}}+6U_{X^{2}Y^{2}}+8U_{X^{3}Y}.$

Thus, $U_{X^{4}}$, $U_{X^{3}Y}$, and $U_{X^{2}Y^{2}}$ are functionally
independent partially normalized invariants. Introducing

$\displaystyle\begin{bmatrix}A_{1}\\\ A_{2}\\\
A_{3}\end{bmatrix}=\begin{bmatrix}1&2&3\\\ 1&4&3\\\
1&2&1\end{bmatrix}\begin{bmatrix}U_{X^{4}}\\\ U_{X^{3}Y}\\\
U_{X^{2}Y^{2}}\end{bmatrix}\\!,$

we have that

$\displaystyle{\rm
d}A_{k}=-\frac{k}{3}A_{k}\mu^{uu}\mod(\omega^{x},\omega^{y}),$

for $k=1,2,3$. We now need to consider the cases

$\displaystyle{\bf H.3.1}\colon\ A_{1}^{2}+A_{2}^{2}+A_{3}^{2}\neq
0,\qquad{\bf H.3.2}\colon\ A_{1}\equiv A_{2}\equiv A_{3}\equiv 0.$

Before considering each case, we note that coordinate expressions for the
invariants $A_{i}$ can be found using the method of recursive moving frame. We
obtained

$\displaystyle
A_{1}=\frac{2\sqrt{|h|}(2C_{2}h_{x}-hC_{2,x})+u_{xy}(C_{2}h_{x}-2hC_{2,x})+u_{xx}(2hC_{2,y}-C_{2}h_{y})}{\sqrt[3]{2}|h|^{7/6}C_{2}^{4/3}a_{33}^{1/3}},$
$\displaystyle
A_{2}=\frac{\sqrt[3]{2}(u_{xx}(hC_{2,y}-2C_{2}h_{y})+(hC_{2,x}-2C_{2}h_{x})(\sqrt{|h|}-u_{xy}))}{u_{xx}|h|^{11/6}(C_{2}a_{33})^{2/3}},$
$\displaystyle
A_{3}=\frac{1}{8h^{3}u_{xx}^{2}a_{33}}\big{(}\sqrt{|h|}u_{xx}(3C_{2}h_{y}-2hC_{2,y})+\sqrt{|h|}u_{xy}(2hC_{2,x}-3C_{2}h_{x})$
$\displaystyle\hphantom{A_{3}=}{}+h(2hC_{2,x}-3C_{2}h_{x})+4h^{2}h_{xx}u_{xx}-h_{y}^{2}u_{xx}^{3}+2h_{x}h_{y}u_{xx}^{2}u_{xy}$
$\displaystyle\hphantom{A_{3}=}{}-h_{x}^{2}u_{xx}u_{xy}^{2}-6hh_{x}^{2}u_{xx}\big{)}.$
(4.7)

####  4.3.1 Case H.3.1

In this case there is $A_{k}$, with $k\in\\{1,2,3\\}$, such that $A_{k}\neq
0$. For the sake of the exposition, assume $A_{3}\neq 0$. The other
possibilities are dealt in a similar fashion. When $A_{3}\neq 0$, one can
normalize $A_{3}=1$. Then

$\displaystyle{\rm
d}A_{1}=\frac{1}{3}\big{[}(3-A_{1})U_{X^{5}}+2(3-A_{1})U_{X^{4}Y}+(9-A_{1})U_{X^{3}Y^{2}}-6-2A_{1}^{2}+A_{1}A_{2}+12A_{2}\big{]}\omega^{x}$
$\displaystyle\hphantom{{\rm
d}A_{1}=}{}+\frac{1}{3}\big{[}(A_{1}\\!-\\!9)U_{X^{5}}\\!+\\!2(A_{1}\\!-\\!12)U_{X^{4}Y}^{\prime}!+\\!(A_{1}\\!-\\!21)U_{X^{3}Y^{2}}\\!-6\\!+\\!2A_{1}^{2}\\!+\\!A_{1}A_{2}\\!-\\!12A_{2}\big{]}\omega^{y},$
$\displaystyle{\rm
d}A_{2}=\frac{1}{6}\big{[}2(3-2A_{2})U_{X^{5}}+8(3-A_{2})U_{X^{4}Y}+2(9-2A_{2})U_{X^{3}Y^{2}}+42-2A_{1}A_{2}+A_{2}^{2}\big{]}\omega^{x}$
$\displaystyle\hphantom{{\rm
d}A_{2}=}{}+\frac{1}{6}\big{[}2(2A_{2}\\!-\\!9)U_{X^{5}}\\!+\\!8(A_{2}\\!-\\!6)U_{X^{4}Y}\\!+\\!2(2A_{2}\\!-\\!15)U_{X^{3}Y^{2}}\\!-\\!42\\!+\\!2A_{1}A_{2}\\!+\\!A_{2}^{2}\big{]}\omega^{y},$

and we have the structure equations

$\displaystyle{\rm
d}\omega^{x}=\frac{1}{12}(8I-2A_{1}+A_{2})\omega^{x}\wedge\omega^{y},\qquad{\rm
d}\omega^{y}=\frac{1}{12}(8I-2A_{1}-A_{2})\omega^{x}\wedge\omega^{y},$

where $I=U_{X^{5}}+2U_{X^{4}Y}+U_{X^{3}Y^{2}}$. Since

$\displaystyle\mathcal{D}_{y}A_{2}+\mathcal{D}_{x}A_{2}=\frac{1}{3}A_{2}^{2}-2I,$

it follows that $I$ can be expressed in terms of $A_{2}$ and its invariant
derivatives. From the syzygy

$\displaystyle
A_{1}(6I-A_{2}^{2})=\frac{A_{2}^{3}}{2}-6IA_{2}^{2}++3A_{2}(4\mathcal{D}_{x}I+2I+4\mathcal{D}_{y}I-5\mathcal{D}_{x}A_{2})$
$\displaystyle\hphantom{A_{1}(6I-A_{2}^{2})=}{}+6\big{(}4I^{2}-3\mathcal{D}_{x}I-9\mathcal{D}_{y}I+3\mathcal{D}_{x}^{2}A_{2}+3\mathcal{D}_{y}\mathcal{D}_{x}A_{2}\big{)},$

it follows that $A_{1}$ can generically be expressed in terms of $A_{2}$ and
its invariant derivatives.

######  Theorem 4.5.

The algebra of differential invariants is generically generated by the single
invariant $A_{2}$.

######  Remark 4.6.

Solving the normalization equation $A_{3}=1$ we obtain

$\displaystyle
a_{33}=\frac{1}{8h^{3}u_{xx}^{2}}\big{(}\sqrt{|h|}u_{xx}(3C_{2}h_{y}-2hC_{2,y})+\sqrt{|h|}u_{xy}(2hC_{2,x}-3C_{2}h_{x})$
$\displaystyle\hphantom{a_{33}=}{}+h(2hC_{2,x}-3C_{2}h_{x})+4h^{2}h_{xx}u_{xx}-h_{y}^{2}u_{xx}^{3}+2h_{x}h_{y}u_{xx}^{2}u_{xy}-h_{x}^{2}u_{xx}u_{xy}^{2}-6hh_{x}^{2}u_{xx}\big{)}.$

Substituting this group parameter normalization into the formula for $A_{2}$
in (4.7) yields the coordinate expression for the generating invariant
$A_{2}$.

####  4.3.2 Case H.3.2

When $A_{1}\equiv A_{2}\equiv A_{3}\equiv 0$, there is no further group
parameter normalizations possible. Then, the structure equations of the
coframe $\\{\omega^{x},\omega^{y},\mu^{uu}\\}$ are

$\displaystyle{\rm
d}\omega^{x}=\frac{1}{2}\mu^{uu}\wedge\omega^{x}+\frac{1}{6}\mu^{uu}\wedge\omega^{y},\qquad{\rm
d}\omega^{y}=\frac{1}{6}\mu^{uu}\wedge\omega^{x}+\frac{1}{2}\mu^{uu}\wedge\omega^{y},\qquad{\rm
d}\mu^{uu}=0.$ (4.8)

##  5 Parabolic points

At a parabolic point, $H=U_{X^{2}}U_{Y^{2}}-U_{XY}^{2}\equiv 0$. Therefore,

$\displaystyle U_{XY}^{2}\equiv U_{X^{2}}U_{Y^{2}}$ (5.1)

and there are now two cases to consider. Namely,

$\displaystyle{\bf P.1}\colon\ U_{X^{2}}\neq 0,\qquad{\bf P.2}\colon\
U_{X^{2}}\equiv 0.$

###  5.1 Case P.1

When $U_{X^{2}}\neq 0$, we can solve for $U_{Y^{2}}$ in (5.1) to obtain

$\displaystyle U_{Y^{2}}\equiv\frac{U_{XY}^{2}}{U_{X^{2}}}.$ (5.2)

Therefore, $U_{X^{2}}$ and $U_{XY}$ are functionally independent partially
normalized invariants. From the recurrence relations (3.3), we conclude that
it is possible to set

$\displaystyle U_{XX}=1,\qquad U_{XY}=0,$ (5.3)

and (5.2) implies that $U_{YY}\equiv 0$. Taking into account the
normalizations (5.3) and the equality $U_{YY}\equiv 0$, the recurrence
relation for $U_{YY}$ in (3.3) implies that

$\displaystyle U_{XY^{2}}\equiv U_{Y^{3}}\equiv 0,$ (5.4)

which in turn yields the recurrence relations

$\displaystyle 0$ $\displaystyle\equiv{\rm
d}U_{XY^{2}}=\big{(}{-}2U_{X^{2}Y}^{2}+U_{X^{2}Y^{2}}\big{)}\omega^{x}+U_{XY^{3}}\omega^{y},$
$\displaystyle 0$ $\displaystyle\equiv{\rm
d}U_{Y^{3}}=U_{XY^{3}}\omega^{x}+U_{Y^{4}}\omega^{y}.$

Therefore,

$\displaystyle U_{Y^{4}}\equiv U_{XY^{3}}\equiv 0,\qquad U_{X^{2}Y^{2}}\equiv
2U_{X^{2}Y}^{2}.$ (5.5)

Considering the recurrence relations for the third order partially normalized
invariants $U_{X^{3}}$ and $U_{X^{2}Y}$, and taking into account the above
constraints on the invariants, we find that

$\displaystyle{\rm
d}U_{X^{3}}=-3\mu^{xu}-\frac{1}{2}U_{X^{3}}\mu^{uu}-3U_{X^{2}Y}\mu^{yx}\mod(\omega^{x},\omega^{y}),$
$\displaystyle{\rm
d}U_{X^{2}Y}=-U_{X^{2}Y}\mu^{yy}\mod(\omega^{x},\omega^{y}).$

From the first equation we conclude that it is possible to normalize
$U_{X^{3}}=0$. As for the second recurrence relation, we have the following
cases to consider

$\displaystyle{\bf P.1.1}\colon\ U_{X^{2}Y}\neq 0,\qquad{\bf P.1.2}\colon\
U_{X^{2}Y}\equiv 0.$

Introducing the ratio $R=\cfrac{u_{xy}}{u_{xx}}$, the coordinate expression
for $U_{X^{2}Y}$ is

$\displaystyle U_{X^{2}Y}=\frac{R_{x}}{Y_{y}-RY_{x}},$

where we recall that $Y_{x}$ and $Y_{y}$ are introduced in (3.4).

####  5.1.1 Case P.1.1

When $U_{X^{2}Y}\neq 0$, we can normalize $U_{X^{2}Y}=1$. Then the recurrence
relations for the non-constant fourth order partially normalized lifted
invariants, i.e., $U_{X^{4}}$ and $U_{X^{3}Y}$, are

$\displaystyle{\rm
d}U_{X^{4}}=-6\mu^{yu}-U_{X^{4}}\mu^{uu}-4U_{X^{3}Y}\mu^{yx}\mod(\omega^{x},\omega^{y}),$
$\displaystyle{\rm
d}U_{X^{3}Y}=-\frac{1}{2}U_{X^{3}Y}\mu^{uu}\mod(\omega^{x},\omega^{y}).$ (5.6)

From the first equation, we see that it is possible to normalize
$U_{X^{4}}=0$. Next, the exterior derivative of the constraints (5.5) yields

$\displaystyle 0={\rm
d}U_{X^{2}Y^{2}}=\big{(}U_{X^{3}Y^{2}}-6U_{X^{3}Y}\big{)}\omega^{x}+(U_{X^{2}Y^{3}}-6)\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{XY^{3}}=\big{(}U_{X^{2}Y^{3}}-6\big{)}\omega^{x}+U_{XY^{4}}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{Y^{4}}=U_{XY^{4}}\omega^{x}+U_{Y^{5}}\omega^{y},$

from which we obtain the following constraints among the order 5 partially
normalized invariants

$\displaystyle U_{Y^{5}}\equiv U_{XY^{4}}\equiv 0,\qquad U_{X^{2}Y^{3}}\equiv
6,\qquad U_{X^{3}Y^{2}}=6U_{X^{3}Y}.$ (5.7)

In light of the second equation in (5.6), we now have to consider the
following cases

$\displaystyle{\bf P.1.1.1}\colon\ U_{X^{3}Y}\neq 0,\qquad{\bf P.1.1.2}\colon\
U_{X^{3}Y}\equiv 0,$

where

$\displaystyle U_{X^{3}Y}=\frac{R_{xx}}{R_{x}\sqrt{|a_{33}u_{xx}}|}.$

####  5.1.2 Case P.1.1.1

When $U_{X^{3}Y}\neq 0$, we set $U_{X^{3}Y}=1$. Then the recurrence relation
for $U_{X^{4}Y}$ is

$\displaystyle{\rm d}U_{X^{4}Y}=-2\mu^{yx}\mod(\omega^{x},\omega^{y}),$

and so we can normalize $U_{X^{4}Y}=0$. At this stage, the recurrence relation
for the only remaining fifth order normalized invariant is

$\displaystyle{\rm
d}U_{X^{5}}=\frac{1}{3}\big{(}3U_{X^{6}}+10U_{X^{5}Y}-36U_{X^{5}}\big{)}\omega^{x}+\frac{1}{9}\big{(}U_{X^{5}Y}+80-63U_{X^{5}}\big{)}\omega^{y}.$
(5.8)

The exterior derivative of the constraints (5.7) yields

$\displaystyle 0\equiv{\rm
d}U_{X^{3}Y^{2}}=\big{(}U_{X^{4}Y^{2}}-6\big{)}\omega^{x}+\big{(}U_{X^{3}Y^{3}}-36\big{)}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{X^{2}Y^{3}}=\big{(}U_{X^{3}Y^{3}}-36\big{)}\omega^{x}+\big{(}U_{X^{2}Y^{4}}-24\big{)}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{XY^{4}}=\big{(}U_{X^{2}Y^{4}}-24\big{)}\omega^{x}+U_{XY^{5}}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{Y^{5}}=U_{XY^{5}}\omega^{x}+U_{Y^{6}}\omega^{y}.$

Thus

$\displaystyle U_{Y^{6}}=0,\qquad U_{XY^{5}}=0,\qquad U_{X^{2}Y^{4}}=24,\qquad
U_{X^{3}Y^{3}}=36,\qquad U_{X^{4}Y^{2}}=6,$

and it follows that $U_{X^{6}}$ and $U_{X^{5}Y}$ are the only functionally
independent invariants of order 6.

From (5.8) we conclude that $U_{X^{5}Y}$ and $U_{X^{6}}$ can be expressed in
terms of $U_{X^{5}}$ and its invariant derivatives. It follows from Theorem
2.7 that $U_{X^{5}}$ generates the algebra of differential invariants.
Introducing the ratios

$\displaystyle
S=\frac{3u_{xx}u_{xxxx}-5u_{xxx}^{2}}{3u_{xx}^{2}}\qquad\text{and}\qquad
L=\frac{u_{xxx}}{u_{xx}},$

we have that

$\displaystyle
U_{X^{5}}=\frac{R_{x}}{36R_{xx}^{4}}\big{(}30LR_{x}R_{xx}R_{xxx}-24LSR_{x}^{2}R_{xx}-5L^{2}R_{x}R_{xx}^{2}-60SR_{x}R_{xx}^{2}-40LR_{xx}^{3}$
$\displaystyle\hphantom{U_{X^{5}}=}{}+120R_{xx}^{2}R_{xxx}-45R_{x}R_{xxx}^{2}+36R_{x}^{2}R_{xx}S_{x}\big{)}.$

Finally, the structure equations of the invariant coframe
$\\{\omega^{x},\omega^{y}\\}$ are

$\displaystyle{\rm d}\omega^{x}=\omega^{y}\wedge\omega^{x},\qquad{\rm
d}\omega^{y}=\frac{1}{3}\omega^{y}\wedge\omega^{x}.$

####  5.1.3 Case P.1.1.2

When $U_{X^{3}Y}\equiv 0$, $0\equiv{\rm
d}U_{X^{3}Y}=U_{X^{4}Y}\omega^{x}+U_{X^{3}Y^{2}}\omega^{y}$, which, when
combined with (5.7), implies that

$\displaystyle U_{Y^{5}}\equiv U_{XY^{4}}\equiv U_{X^{3}Y^{2}}\equiv
U_{X^{4}Y}\equiv 0,\qquad U_{X^{2}Y^{3}}\equiv 6.$ (5.9)

Thus, the recurrence relation for the only non-constant order 5 partially
normalized invariant, namely

$\displaystyle U_{X^{5}}=\frac{T}{3\sqrt{|a_{33}u_{xx}|}^{3}},$

where $T=2LS-3S_{x}$, is

$\displaystyle{\rm
d}U_{X^{5}}=-\frac{3}{2}U_{X^{5}}\mu^{uu}\mod(\omega^{x},\omega^{y}).$ (5.10)

Next, the recurrence relations for the constant invariants (5.9) are

$\displaystyle 0\equiv{\rm
d}U_{X^{4}Y}=\big{(}U_{X^{4}Y}-4U_{X^{5}}\big{)}\omega^{x}+U_{X^{4}Y^{2}}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{X^{3}Y^{2}}=U_{X^{4}Y^{2}}\omega^{x}+U_{X^{3}Y^{3}}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{X^{2}Y^{3}}=U_{X^{3}Y^{3}}\omega^{x}+\big{(}U_{X^{2}Y^{4}}-24\big{)}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{XY^{4}}=\big{(}U_{X^{2}Y^{4}}-24\big{)}\omega^{x}+U_{XY^{5}}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{Y^{5}}=U_{XY^{5}}\omega^{x}+U_{Y^{6}}\omega^{y}.$

These equations imply that

$\displaystyle U_{Y^{6}}\equiv U_{XY^{5}}\equiv U_{X^{3}Y^{3}}\equiv
U_{X^{4}Y^{2}}\equiv 0,\qquad U_{X^{2}Y^{4}}\equiv 24,\qquad U_{X^{5}Y}\equiv
4U_{X^{5}}.$ (5.11)

In light of (5.10), we have the following cases to consider

$\displaystyle{\bf P.1.1.2.1}\colon\ U_{X^{5}}\neq 0,\qquad{\bf
P.1.1.2.2}\colon\ U_{X^{5}}\equiv 0.$

####  5.1.4 Case P.1.1.2.1

In this case we normalize $U_{X^{5}}=1$. Then the recurrence relation for
$U_{X^{6}}$ is

$\displaystyle{\rm d}U_{X^{6}}=-3\mu^{yx}\mod(\omega^{x},\omega^{y}),$

and it is therefore also possible to set $U_{X^{6}}=0$. At this stage all
invariants of order 6 or less are constant and the only non-constant invariant
of order 7 is

$\displaystyle
U_{X^{7}}=-\frac{1}{6}-\frac{3^{2/3}L^{2}}{2T^{2/3}}-\frac{3^{5/3}S}{2T^{2/3}}-\frac{7T_{x}^{2}}{2\cdot
3^{1/3}T^{8/3}}+\frac{3^{2/3}T_{xx}}{T^{5/3}}.$

Similarly, the only non-phantom invariant of order 8 is $U_{X^{8}}$. From the
recurrence relation

$\displaystyle{\rm
d}U_{X^{7}}=\bigg{(}U_{X^{8}}-\frac{35}{2}\bigg{)}\omega^{x}$

it follows that $U_{X^{8}}=\mathcal{D}_{x}U_{X^{7}}+\frac{35}{2}$, and from
Theorem 2.7, $U_{X^{7}}$ generates the algebra of differential invariants.
Finally, the structure equations are

$\displaystyle{\rm d}\omega^{x}=0,\qquad{\rm
d}\omega^{y}=\frac{5}{3}\omega^{y}\wedge\omega^{x}.$

####  5.1.5 Case P.1.1.2.2

When $U_{X^{5}}\equiv 0$, we have that

$\displaystyle 0\equiv{\rm d}U_{X^{5}}=U_{X^{6}}\omega^{x},$

which when combined with (5.11), implies that all sixth order invariants are
constant. Similarly, all higher order invariants are constant and there are no
further possible normalizations. Finally, the structure equations for the
coframe $\\{\omega^{x},\omega^{y},\mu^{yx},\mu^{uu}\\}$ are

$\displaystyle{\rm
d}\omega^{x}=\frac{1}{2}\omega^{x}\wedge\omega^{y},\qquad{\rm
d}\omega^{y}=\mu^{yx}\wedge\omega^{x},$ $\displaystyle{\rm
d}\mu^{yx}=\frac{1}{2}\mu^{yx}\wedge\omega^{y}+\frac{1}{2}\mu^{uu}\wedge\mu^{yx},\qquad{\rm
d}\mu^{uu}=\mu^{yx}\wedge\omega^{x}.$

####  5.1.6 Case P.1.2

When $U_{X^{2}Y}\equiv 0$, we have $U_{X^{2}Y}\equiv U_{XY^{2}}\equiv
U_{Y^{3}}\equiv 0$, in light of (5.4). We also recall that $U_{X^{3}}$ is
normalized to zero. From the recurrence relations

$\displaystyle 0\equiv{\rm
d}U_{X^{2}Y}=U_{X^{3}Y}\omega^{x}+U_{X^{2}Y^{2}}\omega^{y},$ $\displaystyle
0\equiv{\rm d}U_{XY^{2}}=U_{X^{2}Y^{2}}\omega^{x}+U_{XY^{3}}\omega^{y},$
$\displaystyle 0\equiv{\rm
d}U_{Y^{3}}=U_{XY^{3}}\omega^{x}+U_{Y^{4}}\omega^{y},$

we conclude that

$\displaystyle U_{XY^{3}}\equiv U_{X^{2}Y^{2}}\equiv U_{X^{3}Y}\equiv
U_{Y^{4}}\equiv 0.$ (5.12)

Thus,

$\displaystyle U_{X^{4}}=\frac{S}{a_{33}u_{xx}}$

is the lowest order non-zero invariant and the recurrence relation

$\displaystyle{\rm d}U_{X^{4}}=-U_{X^{4}}\mu^{uu}\mod(\omega^{x},\omega^{y}),$

leads us to consider the following cases

$\displaystyle{\bf P.1.2.1}\colon\ U_{X^{4}}\neq 0,\qquad{\bf P.1.2.2}\colon\
U_{X^{4}}\equiv 0.$

####  5.1.7 Case P.1.2.1

In this case we can normalize $U_{X^{4}}=1$. Since the recurrence relations
for the vanishing invariants (5.12) are of the form $0\equiv{\rm
d}U_{J}=U_{J,i}\omega^{i}$, all fifth order partially normalized invariants
are zero except for $U_{X^{5}}$. Similarly, all sixth order partially
normalized invariants are zero except for $U_{X^{6}}$. Since

$\displaystyle{\rm
d}U_{X^{5}}=-\frac{1}{2}\big{(}10+3U_{X^{5}}^{2}-2U_{X^{6}}\big{)}\omega^{x},$

the function $U_{X^{5}}$ is a _genuine_ differential invariant not depending
on the remaining group parameters. By a similar argument, we see that for
$k\geq 5$, $U_{X^{k}}$ are genuine differential invariants, while
$U_{X^{k}Y^{\ell}}\equiv 0$ for $\ell>0$ and $k+\ell\geq 4$. It follows that

$\displaystyle U_{X^{5}}=\frac{3T}{S^{3/2}}$

generates the algebra of differential invariants.

When $U_{X^{5}}=c$ is constant, it follows that $U_{X^{k}}$, $k\geq 5$, are
all constant and the symmetry group of these surfaces has structure equations

$\displaystyle{\rm d}\omega^{x}=0,\qquad{\rm
d}\omega^{y}=\mu^{yx}\wedge\omega^{x}+\mu^{yy}\wedge\omega^{y},$
$\displaystyle{\rm
d}\mu^{yu}=c\omega^{x}\wedge\mu^{yu}+\frac{1}{3}\omega^{x}\wedge\mu^{yx}+\mu^{yu}\wedge\mu^{yy},$
$\displaystyle{\rm
d}\mu^{yx}=\frac{c}{2}\omega^{x}\wedge\mu^{yx}+\mu^{yu}\wedge\omega^{x}+\mu^{yx}\wedge\mu^{yy},\qquad{\rm
d}\mu^{yy}=0.$

####  5.1.8 Case P.1.2.2

If $U_{X^{4}}\equiv 0$, then in light of (5.12) all fourth order partially
normalized invariants are zero and there are no non-trivial invariants. These
surfaces have a symmetry group with structure equations

$\displaystyle{\rm d}\omega^{x}=\frac{1}{2}\mu^{uu}\wedge\omega^{x},\qquad{\rm
d}\omega^{y}=\mu^{yx}\wedge\omega^{x}+\mu^{yy}\wedge\omega^{y},\qquad{\rm
d}\mu^{uu}=0,\qquad{\rm d}\mu^{yy}=0,$ $\displaystyle{\rm
d}\mu^{yu}=\mu^{yu}\wedge\mu^{yy}+\mu^{uu}\wedge\mu^{yu},\qquad{\rm
d}\mu^{yx}=\frac{1}{2}\mu^{uu}\wedge\mu^{yx}+\mu^{yx}\wedge\mu^{yy}+\mu^{yu}\wedge\omega^{x}.$

###  5.2 Case P.2

If $U_{X^{2}}\equiv 0$, then equation (5.1) implies that $U_{XY}\equiv 0$.
Since

$\displaystyle 0={\rm
d}U_{XY}=-U_{YY}\mu^{yx}\mod(\omega^{x}\wedge\omega^{y}),$

it follows that $U_{YY}\equiv 0$. Such surfaces have a 9-dimensional symmetry
group with structure equations

$\displaystyle{\rm
d}\omega^{x}=\mu^{xx}\wedge\omega^{x}+\mu^{xy}\wedge\omega^{y},\qquad{\rm
d}\omega^{y}=\mu^{yx}\wedge\omega^{x}+\mu^{yy}\wedge\omega^{y},\qquad{\rm
d}\mu^{uu}=0,$ $\displaystyle{\rm d}\mu^{xx}=\mu^{yx}\wedge\mu^{xy},\qquad{\rm
d}\mu^{xy}=\mu^{xy}\wedge\mu^{xx}+\mu^{yy}\wedge\mu^{xy},$ $\displaystyle{\rm
d}\mu^{xu}=\mu^{xu}\wedge\mu^{xx}+\mu^{yu}\wedge\mu^{xy}+\mu^{uu}\wedge\mu^{xu},\qquad{\rm
d}\mu^{yx}=\mu^{xx}\wedge\mu^{yx}+\mu^{yx}\wedge\mu^{yy},$ $\displaystyle{\rm
d}\mu^{yy}=\mu^{xy}\wedge\mu^{yx},\qquad{\rm
d}\mu^{yu}=\mu^{xu}\wedge\mu^{yx}+\mu^{yu}\wedge\mu^{yy}+\mu^{uu}\wedge\mu^{yu}.$

##  6 Homogeneous surfaces

As mentioned in the introduction, differential geometers have been especially
interested in the study of homogeneous surfaces that arise from the
equivalence problem [1, 5, 7]. These surfaces are characterized by the
property that all relative and differential invariants are constant.
Therefore, homogeneous surfaces are described as solutions to certain systems
of partial differential equations. We now consider several examples, with the
understanding that it is not our intention to recover the full classifications
found in [1, 5, 7].

######  Example 6.1.

As our first example, let us consider the branch EH.2.1. Surfaces belonging to
this branch satisfy the partial differential equations $U_{X^{3}}\equiv
U_{Y^{3}}\equiv 0$ and the non-degeneracy condition $U_{X^{2}Y^{2}}\neq 0$. In
jet coordinates, these conditions translate to the formulas

$\displaystyle
u_{xyy}=\frac{u_{xxx}\big{(}u_{xx}u_{yy}-4u_{xy}^{2}\big{)}+6u_{xx}u_{xy}u_{xxy}}{3u_{xx}^{2}},\qquad
u_{yyy}=\frac{u_{yy}(3u_{xx}u_{xxy}-2u_{xxx}u_{xy})}{u_{xx}^{2}},$ (6.1)

and

$\displaystyle\big{(}4u_{xxx}^{2}\\!+3u_{xx}u_{xxxx}\big{)}u_{xy}^{2}-18u_{xx}u_{xy}u_{xxx}u_{xxy}\\!+9u_{xx}^{2}u_{xxy}^{2}\\!+u_{xx}u_{yy}\big{(}5u_{xxx}^{2}\\!-3u_{xx}u_{xxxx}\big{)}\neq
0.$

Our results say that all surfaces satisfying this system are equivalent, and
each is a homogeneous space with symmetry group of dimension 3. A normal form
for this branch can therefore be taken as any solution to the above system.
Completing (6.1) to an involutive system [28], one obtains a _maximally
overdetermined_ fifth order system, which can be solved using the Frobenius
theorem. We find that the non-degenerate quadrics

$\displaystyle\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\pm\frac{u^{2}}{c^{2}}=1,\qquad\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{u^{2}}{c^{2}}=-1,$

with $c\neq 0$, satisfy the constraints of this branch. These surfaces
correspond to case (1) in [1, Theorem 1.1].

######  Example 6.2.

In Case EH.2.2 we also obtain homogeneous surfaces since all the invariants
are constant. In this case, the surface must be a solution to the system of
differential equations

$\displaystyle
u_{xyy}=\frac{u_{xxx}\big{(}u_{xx}u_{yy}-4u_{xy}^{2}\big{)}+6u_{xx}u_{xy}u_{xxy}}{3u_{xx}^{2}},\qquad
u_{yyy}=\frac{u_{yy}(3u_{xx}u_{xxy}-2u_{xxx}u_{xy})}{u_{xx}^{2}},$
$\displaystyle\big{(}4u_{xxx}^{2}\\!+3u_{xx}u_{xxxx}\big{)}u_{xy}^{2}\\!-18u_{xx}u_{xy}u_{xxx}u_{xxy}\\!+9u_{xx}^{2}u_{xxy}^{2}\\!+u_{xx}u_{yy}\big{(}5u_{xxx}^{2}\\!-3u_{xx}u_{xxxx}\big{)}=0.$

One can verify that the remain two non-degenerate quadrics
$u=\frac{y^{2}}{b^{2}}\pm\frac{x^{2}}{a^{2}}$ are solutions. Setting $a=b=1$,
we recover case (8) of [5, Theorem 1] with $\alpha=2$.

######  Example 6.3.

In Case P.1.1.2.2, the homogeneous surface must satisfy the system of
differential equations

$\displaystyle u_{xx}u_{yy}-u_{xy}^{2}=0,\qquad
45u_{xx}u_{xxx}u_{xxxx}-9u_{xx}^{2}u_{xxxxx}-40u_{xxx}^{3}=0,$ $\displaystyle
u_{xx}^{2}u_{xxxy}-u_{xx}u_{xy}u_{xxxx}-2u_{xx}u_{xxy}u_{xxx}+2u_{xy}u_{xxx}^{2}=0,$

and the non-degeneracy conditions

$\displaystyle u_{xx}\neq 0,\qquad u_{xx}u_{xxy}-u_{xy}u_{xxx}\neq 0.$

A solution is given by $u=x^{2}y^{-1}$, corresponding to case (1) of [5,
Theorem 1] with $\alpha=2$ and $\beta=-1$.

######  Example 6.4.

A homogeneous surface in branch P.1.2.2 must satisfy the system of partial
differential equations

$\displaystyle u_{xx}u_{yy}-u_{xy}^{2}=0,\qquad
u_{xxy}u_{xx}-u_{xy}u_{xxx}=0,\qquad 3u_{xx}u_{xxxx}-5u_{xxx}^{2}=0.$

A solution is given by $u=x^{2}$, corresponding to case (1) of [5, Theorem 1].

######  Example 6.5.

A homogeneous surface in branch P.2 will be a solution to the system of
differential equations

$\displaystyle u_{xx}=u_{xy}=u_{yy}=0.$

The general solution being a plane $u=ax+by+c$.

The above examples show that attempting to recover the homogeneous surfaces
from the systems of partial differential equations one obtains by setting the
relative or differential invariants to constant values can be extremely
challenging as these equations are highly nonlinear and of high order.
Luckily, it is possible to avoid these difficulties by integrating the moving
frame equations instead [10]. To see how this works, let
$\widehat{\rho}^{(n)}=\big{(}\rho^{(n)},z^{(n)}\big{)}$ be a partial right
moving frame. As is customary, we also refer to $\rho^{(n)}=\rho^{(n)}\in G$
as a partial right moving frame. Then let
$\overline{\rho}^{(n)}=\big{(}\rho^{(n)}\big{)}^{-1}$ denote the partial left
moving frame. Taking the exterior derivative of the identity
$\rho^{(n)}\overline{\rho}^{(n)}=\mathds{1}$, we find that

$\displaystyle{\rm
d}\overline{\rho}^{(n)}=-\overline{\rho}^{(n)}\bm{\mu}^{*},$ (6.2)

where $\bm{\mu}^{*}=\big{(}\rho^{(n)}\big{)}^{*}\bm{\mu}$ denotes the right
moving frame pull-back of the Maurer–Cartan forms. To proceed further, let

$\displaystyle\overline{\rho}^{(n)}=\begin{bmatrix}E&z\\\
0&1\end{bmatrix}\\!,$

where $E=(\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3})\in GL(3)$ is a frame on
the homogeneous surface $S\subset\mathbb{R}^{3}$ and $z\in\mathbb{R}^{3}$ is a
point in $S$. It therefore follows that if one can integrate the moving frame
equation (6.2) for $\overline{\rho}^{(n)}$, a parametrization of the
homogeneous surface will be given by the vector $z\in\mathbb{R}^{3}$. We now
show how this works with a concrete example.

######  Example 6.6.

In this example we will deduce the homogeneous surface that originates from
Case H.3.2. Recall the structure equations obtained in (4.8). From the third
equation, it follows that, locally,

$\displaystyle\mu^{uu}={\rm d}a.$

Next, introduce

$\displaystyle\overline{\omega}^{x}={\rm
e}^{-2a/3}(\omega^{x}+\omega^{y}),\qquad\overline{\omega}^{y}={\rm
e}^{-a/3}(\omega^{x}-\omega^{y}).$ (6.3)

Using (4.8) we find that

$\displaystyle{\rm d}\overline{\omega}^{x}={\rm d}\overline{\omega}^{y}=0.$

Therefore, locally,

$\displaystyle\overline{\omega}^{x}=2{\rm
d}\overline{x},\qquad\overline{\omega}=2{\rm d}\overline{y},$ (6.4)

for certain functions $\overline{x}$ and $\overline{y}$. Substituting (6.4) in
(6.3) and solving for $\omega^{x}$ and $\omega^{y}$ we obtain

$\displaystyle\omega^{x}={\rm e}^{2a/3}{\rm d}\overline{x}+{\rm e}^{a/3}{\rm
d}\overline{y},\qquad\omega^{y}={\rm e}^{2a/3}{\rm d}\overline{x}-{\rm
e}^{a/3}{\rm d}\overline{y}.$

Since

$\displaystyle\bm{\mu}^{*}=\begin{bmatrix}\frac{1}{2}(\omega^{x}-\omega^{y})+\frac{1}{2}\mu^{uu}&-\frac{1}{2}(\omega^{x}-\omega^{y})+\frac{1}{6}\mu^{uu}&0&-\omega^{x}\\\\[4.30554pt]
\frac{1}{2}(\omega^{x}-\omega^{y})+\frac{1}{6}\mu^{uu}&-\frac{1}{2}(\omega^{x}-\omega^{y})+\frac{1}{2}\mu^{uu}&0&-\omega^{y}\\\
-\omega^{x}&\omega^{y}&\mu^{uu}&0\\\ 0&0&0&0\end{bmatrix}$
$\displaystyle\hphantom{\bm{\mu}^{*}}=\begin{bmatrix}{\rm e}^{a/3}{\rm
d}\overline{y}+\frac{1}{2}{\rm d}a&-a^{a/3}{\rm d}\overline{y}+\frac{1}{6}{\rm
d}a&0&-{\rm e}^{2a/3}{\rm d}\overline{x}-{\rm e}^{a/3}{\rm
d}\overline{y}\\\\[4.30554pt] {\rm e}^{a/3}{\rm d}\overline{y}+\frac{1}{6}{\rm
d}a&-{\rm e}^{a/3}{\rm d}\overline{y}+\frac{1}{2}{\rm d}a&0&-{\rm
e}^{2a/3}{\rm d}\overline{x}+{\rm e}^{a/3}{\rm d}\overline{y}\\\\[4.30554pt]
-{\rm e}^{2a/3}{\rm d}\overline{x}-{\rm e}^{a/3}{\rm d}\overline{y}&{\rm
e}^{2a/3}{\rm d}\overline{x}-{\rm e}^{a/3}{\rm d}\overline{y}&{\rm d}a&0\\\
0&0&0&0\end{bmatrix}\\!,$ (6.5)

equation (6.2) yields

$\displaystyle{\rm d}z=\big{(}{\rm e}^{2a/3}{\rm d}\overline{x}+{\rm
e}^{a/3}{\rm d}\overline{y}\big{)}\mathbf{e}_{1}+\big{(}{\rm e}^{2a/3}{\rm
d}\overline{x}-{\rm e}^{a/3}{\rm d}\overline{y}\big{)}\mathbf{e}_{2},$
$\displaystyle{\rm d}\mathbf{e}_{1}=-\bigg{(}{\rm e}^{a/3}{\rm
d}\overline{y}+\frac{{\rm d}a}{2}\bigg{)}\mathbf{e}_{1}-\bigg{(}{\rm
e}^{a/3}{\rm d}\overline{y}+\frac{{\rm
d}a}{2}\bigg{)}\mathbf{e}_{2}+\big{(}{\rm e}^{2a/3}{\rm d}\overline{x}+{\rm
e}^{a/3}{\rm d}\overline{y}\big{)}\mathbf{e}_{3},$ $\displaystyle{\rm
d}\mathbf{e}_{2}=-\bigg{(}{-}{\rm e}^{a/3}{\rm d}\overline{y}+\frac{{\rm
d}a}{6}\bigg{)}\mathbf{e}_{1}-\bigg{(}{-}{\rm e}^{a/3}{\rm
d}\overline{y}+\frac{{\rm d}a}{2}\bigg{)}\mathbf{e}_{2}-\big{(}{\rm
e}^{2a/3}{\rm d}\overline{x}-{\rm e}^{a/3}{\rm
d}\overline{y}\big{)}\mathbf{e}_{3},$ $\displaystyle{\rm
d}\mathbf{e}_{3}=-{\rm d}a\mathbf{e}_{3},$

from which we conclude that

$\displaystyle z_{\overline{x}}={\rm
e}^{2a/3}(\mathbf{e}_{1}+\mathbf{e}_{2}),\qquad z_{\overline{y}}={\rm
e}^{a/3}(\mathbf{e}_{1}-\mathbf{e}_{2}),$
$\displaystyle\mathbf{e}_{1,\overline{x}}={\rm
e}^{2a/3}\mathbf{e}_{3},\qquad\mathbf{e}_{1,\overline{y}}={\rm
e}^{a/3}(-\mathbf{e}_{1}-\mathbf{e}_{2}+\mathbf{e}_{3}),\qquad\mathbf{e}_{1,a}=-\frac{1}{2}\mathbf{e}_{1}-\frac{1}{6}\mathbf{e}_{2},$
$\displaystyle\mathbf{e}_{2,\overline{x}}=-{\rm
e}^{2a/3}\mathbf{e}_{3},\qquad\mathbf{e}_{2,\overline{y}}={\rm
e}^{a/3}(\mathbf{e}_{1}+\mathbf{e}_{2}+\mathbf{e}_{3}),\qquad\mathbf{e}_{2,a}=-\frac{1}{6}\mathbf{e}_{1}-\frac{1}{2}\mathbf{e}_{2},$
$\displaystyle\mathbf{e}_{3,\overline{x}}=\mathbf{e}_{3,\overline{y}}=0,\qquad\mathbf{e}_{3,a}=-\mathbf{e}_{3}.$

and

$\displaystyle z_{\overline{x}\overline{x}}=0,\qquad
z_{\overline{x}\overline{y}}=2e^{a}\mathbf{e}_{3},\qquad
z_{\overline{y}\overline{y}}=-2{\rm
e}^{2a/3}(\mathbf{e}_{1}+\mathbf{e}_{2}),\qquad
z_{\overline{y}\overline{y}\overline{y}}=-4e^{a}\mathbf{e}_{3}.$

Integrating the latter system of equations, we obtain

$\displaystyle z=\big{(}{\rm e}^{2a/3}\overline{x}+{\rm
e}^{a/3}\overline{y}+{\rm
e}^{2a/3}\overline{y}^{2}\big{)}\mathbf{e}_{1}+\big{(}{\rm
e}^{2a/3}\overline{x}-{\rm e}^{a/3}\overline{y}+{\rm
e}^{2a/3}\overline{y}^{2}\big{)}\mathbf{e}_{2}-2e^{a}\bigg{(}\overline{x}\overline{y}+\frac{1}{3}\overline{y}^{3}\bigg{)}\mathbf{e}_{3}.$

Introducing the variables

$\displaystyle x={\rm e}^{2a/3}\overline{x}+{\rm e}^{a/3}\overline{y}+{\rm
e}^{2a/3}\overline{y}^{2},\quad\ \ y={\rm e}^{2a/3}\overline{x}-{\rm
e}^{a/3}\overline{y}+{\rm e}^{2a/3}\overline{y}^{2},\quad\ \
u=-2e^{a}\bigg{(}\overline{x}\overline{y}+\frac{1}{3}\overline{y}^{3}\bigg{)},$

we find that

$\displaystyle
u=-\frac{x^{2}}{2}+\frac{y^{2}}{2}-\frac{x^{3}}{6}+\frac{x^{2}y}{2}-\frac{xy^{2}}{2}+\frac{y^{3}}{6}.$

Under the change of variables $(x,y,u)\to(-x,-y,-u)$ we get

$\displaystyle
u=\frac{x^{2}}{2}-\frac{y^{2}}{2}+\frac{x^{3}}{6}-\frac{x^{2}y}{2}+\frac{xy^{2}}{2}-\frac{y^{3}}{6}.$
(6.6)

This surface is equivalent to the Cayley surface
$u(\widetilde{x},\widetilde{y})=\widetilde{x}\widetilde{y}-\frac{1}{3}\widetilde{x}^{3}$
[19], under the change of variables
$\widetilde{x}=\frac{1}{\sqrt[3]{2}}(y-x)$,
$\widetilde{y}=-\frac{1}{\sqrt[3]{2^{2}}}(x+y)$.

Finally, one can also verify that (6.6) is a solution to the system of partial
differential equations

$\displaystyle C_{1}+C_{2}\sqrt{|h|}=A_{1}=A_{2}=A_{3}=0,$

which the surface must satisfy to be in branch H.3.2.

######  Remark 6.7.

We note how the recurrence formula and the equivariant moving frame calculus
facilitated the above computations by providing us with the matrix
$\bm{\mu}^{*}$ in (6.5) essentially for free.

##  7 Result summary

In this section we summarize the results obtained in this paper by listing the
normal forms of surfaces, given as graphs of functions $u(x,y)$, for the
different, suitably generic, branches of the equivalence problem we considered
in this paper. We also provide the possible dimensions of the self-symmetry
group and recall the branches whose differential invariant algebra is
generated by a single invariant. Note that we do not identify all possible
equivalence classes. For homogeneous surfaces, this would require a thorough
inspection of all possible constant values that differential invariant can
take. For surfaces admitting non-trivial invariants, this would require a
detailed analysis of the signature manifold [20]. Throughout, $\epsilon=\pm
1$, with $\epsilon=1$ for elliptic points and $\epsilon=-1$ for hyperbolic
points.

Case EH.1:

$\displaystyle
u(x,y)=\frac{1}{2}x^{2}+\epsilon\frac{1}{2}y^{2}+\frac{1}{6}x^{3}+\epsilon\frac{1}{2}x^{2}y+\sum_{i,j\geq
0}c_{i(4+j)}\frac{1}{i!(4+j)!}x^{i}y^{4+j}+\sum_{\begin{subarray}{c}i+j\geq
4\\\ j<4\end{subarray}}F_{ij}(\mathbf{c})\frac{1}{i!j!}x^{i}y^{j},$

where $\mathbf{c}$ is the infinite vector of coefficients $c_{i(4+j)}$,
$i,j\geq 0$ and $F_{ij}$ are certain universal, determinable, functions
thereof. These surfaces have self-symmetry groups of dimension 0, 1 or 2,
depending on the particularities of $\mathbf{c}$. Also, the algebra of
differential invariants is generated by a single fourth order invariant.

Case EH.2.1:

$\displaystyle
u(x,y)=\frac{1}{2}x^{2}+\epsilon\frac{1}{2}y^{2}+\frac{3\epsilon}{4!}x^{4}+\frac{1}{4}x^{2}y^{2}+\frac{3\epsilon}{4!}y^{4}+\text{h.o.t.},$

where h.o.t. are higher order terms. These surfaces have self-symmetry group
of dimension 3, and there are no differential invariants.

Case H.2.2:

$\displaystyle u(x,y)=\frac{1}{2}x^{2}+\epsilon\frac{1}{2}y^{2}.$

These surfaces have self-symmetry groups of dimension 4, and there are no
differential invariants.

Case H.3.1:

$\displaystyle
u(x,y)=\frac{1}{2}x^{2}-\frac{1}{2}y^{2}+\frac{1}{6}x^{3}-\frac{1}{2}x^{2}y-\tilde{\epsilon}\frac{1}{2}xy^{2}+\tilde{\epsilon}\frac{1}{6}y^{3}$
$\displaystyle\hphantom{u(x,y)=}{}+\sum_{i,j\geq
0}c_{(2+i)(2+j)}\frac{1}{(2+i)!(2+j)!}x^{2+i}y^{2+j}+\sum_{\begin{subarray}{c}i+j\geq
4\\\ j<2~{}\text{or
}i<2\end{subarray}}F_{ij}(\mathbf{c})\frac{1}{i!j!}x^{i}y^{j},$

where $\mathbf{c}$ is the infinite vector of $c_{(2+i)(2+j)}$, $i,j\geq 0$,
$c_{22}\neq 0$, and $F_{ij}$ are certain universal, determinable, functions
thereof. These surfaces have self-symmetry groups of dimension 0, 1 or 2,
depending on the particularities of $\mathbf{c}$. Furthermore, the algebra of
differential invariants is generated by a single fourth order invariant.

Case H.3.2:

$\displaystyle
u(x,y)=\frac{1}{2}x^{2}-\frac{1}{2}y^{2}+\frac{1}{6}x^{3}-\frac{1}{2}x^{2}y+\frac{1}{2}xy^{2}-\frac{1}{6}y^{3}.$

The self-symmetry group has dimension 3, and there are no differential
invariants.

Case P1.1.1:

$\displaystyle
u(x,y)=\frac{1}{2}x^{2}+\frac{1}{2}x^{2}y+\frac{1}{6}x^{3}y+\frac{1}{2}x^{2}y^{2}+\sum_{i,j\geq
0}c_{(5+i)j}\frac{1}{(5+i)!j!}x^{5+i}y^{j}+\\!\\!\sum_{\begin{subarray}{c}i+j\geq
5\\\ i<5\end{subarray}}\\!F_{ij}(\mathbf{c})\frac{1}{i!j!}x^{i}y^{j},$

where $\mathbf{c}$ is the infinite vector of $c_{(5+i)(j)}$, $i,j\geq 0$ and
$F_{ij}$ are certain universal, determinable, functions thereof. These
surfaces have self-symmetry groups of dimension 0, 1 or 2, depending on the
particularities of $\mathbf{c}$. In this case, the algebra of differential
invariants is generated by a fifth order invariant.

Case P.1.1.2.1:

$\displaystyle
u(x,y)=\frac{1}{2}x^{2}+\frac{1}{2}x^{2}y+\frac{1}{2}x^{2}y^{2}+\frac{1}{5!}x^{5}+\frac{1}{2}x^{2}y^{3}+\frac{1}{2}x^{2}y^{4}+\frac{1}{30}x^{5}y+\text{h.o.t.},$

where h.o.t. are higher order terms. The self-symmetry group has dimension 2,
and the differential invariant algebra is generated by a seventh order
invariant.

Case P.1.1.2.2:

$\displaystyle
u(x,y)=\frac{1}{2}x^{2}+\frac{1}{2}x^{2}y+\frac{1}{2}x^{2}y^{2}+\frac{1}{5!}x^{5}+\frac{1}{2}x^{2}y^{3}+\frac{1}{2}x^{2}y^{4}+\text{h.o.t.},$

where h.o.t. are higher order terms. The self-symmetry group has dimension 4,
and there are no differential invariants.

Case P.1.2.1:

$\displaystyle u(x,y)=\frac{1}{2}x^{2}+\frac{1}{4!}x^{4}+\sum_{i\geq
0}c_{(5+i)0}\frac{1}{(5+i)!}x^{5+i}.$

The self-symmetry group has dimension 3, 4 or 5 depending on the series of
$c_{(5+i)0}$, and the invariant differential algebra is generated by a fifth
invariant.

Case P.1.2.2:

$\displaystyle u(x,y)=\frac{1}{2}x^{2}.$

The symmetry group has dimension 6, and there are no differential invariants.

Case P.2:

$\displaystyle u(x,y)=0$

has a 9-dimensional self-symmetry group, and there are no differential
invariants.

### Acknowledgement

We would like to thank the referees for their valuable comments, which helped
improve the exposition of the paper.

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