Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

Contractions of Degenerate Quadratic Algebras,
Abstract and Geometric

Mauricio A. ESCOBAR RUIZ †, Willard MILLER Jr. ‡ and Eyal SUBAG §

M.A. Escobar Ruiz, W. Miller, Jr. and E. Subag

† Centre de Recherches Mathématiques, Université de Montreal,
† C.P. 6128, succ. Centre-Ville, Montréal, QC H3C 3J7, Canada
<EMAIL_ADDRESS>
‡ School of Mathematics, University of Minnesota, Minneapolis, Minnesota,
55455, USA<EMAIL_ADDRESS>https://www.ima.umn.edu/~miller/ § Department of
Mathematics, Pennsylvania State University, State College,
§ Pennsylvania, 16802 USA<EMAIL_ADDRESS>
Received August 09, 2017, in final form December 26, 2017; Published online
December 31, 2017

Quadratic algebras are generalizations of Lie algebras which include the
symmetry algebras of 2nd order superintegrable systems in 2 dimensions as
special cases. The superintegrable systems are exactly solvable physical
systems in classical and quantum mechanics. Distinct superintegrable systems
and their quadratic algebras can be related by geometric contractions, induced
by Bôcher contractions of the conformal Lie algebra
$\mathfrak{so}(4,\mathbb{C})$ to itself. In 2 dimensions there are two kinds
of quadratic algebras, nondegenerate and degenerate. In the geometric case
these correspond to 3 parameter and 1 parameter potentials, respectively. In a
previous paper we classified all abstract parameter-free nondegenerate
quadratic algebras in terms of canonical forms and determined which of these
can be realized as quadratic algebras of 2D nondegenerate superintegrable
systems on constant curvature spaces and Darboux spaces, and studied the
relationship between Bôcher contractions of these systems and abstract
contractions of the free quadratic algebras. Here we carry out an analogous
study of abstract parameter-free degenerate quadratic algebras and their
possible geometric realizations. We show that the only free degenerate
quadratic algebras that can be constructed in phase space are those that arise
from superintegrability. We classify all Bôcher contractions relating
degenerate superintegrable systems and, separately, all abstract contractions
relating free degenerate quadratic algebras. We point out the few exceptions
where abstract contractions cannot be realized by the geometric Bôcher
contractions.

Bôcher contractions; quadratic algebras; superintegrable systems; conformal
superintegrability; Poisson structures

22E70; 16G99; 37J35; 37K10; 33C45; 17B60; 81R05; 33C45

##  1 Introduction

An abstract degenerate (quantum) quadratic algebra $Q$ is a noncommutative
multiparameter associative algebra generated by linearly independent operators
$X$, $H$, $L_{1}$, $L_{2}$, with parameters $a_{i}$, such that $H$ is in the
center and the following commutation relations hold [17]:

$\displaystyle[X,L_{j}]=\sum_{0\leq e_{1}+e_{2}+e_{3}+e_{4}\leq
1}P^{(j)}_{e_{1},e_{2},e_{3},e_{4}}L_{1}^{e_{1}}L_{2}^{e_{2}}H^{e_{3}}X^{2e_{4}},\qquad
j=1,2,$ (1.1) $\displaystyle[L_{1},L_{2}]=\sum_{0\leq
e_{1}+e_{2}+e_{3}+e_{4}\leq
1}T_{e_{1},e_{2},e_{3},e_{4}}\\{L_{1}^{e_{1}},L_{2}^{e_{2}},X\\}H^{e_{3}}X^{2e_{4}}.$
(1.2)

Finally, there is the relation:

$\displaystyle G\equiv\sum_{0\leq e_{1}+e_{2}+e_{3}+e_{4}\leq
2}S_{e_{1},e_{2},e_{3},e_{4}}\big{\\{}L_{1}^{e_{1}},L_{2}^{e_{2}},X^{2e_{4}}\big{\\}}H^{e_{3}}+c_{1}XL_{1}X+c_{2}XL_{2}X=0,$
(1.3) $\displaystyle X^{0}=H^{0}=I,$

where $\\{L_{1}^{e_{1}},L_{2}^{e_{2}},X^{2e_{4}}\\}$ is the 6-term symmetrizer
of three operators. The constants $P^{(j)}_{e_{1},e_{2},e_{3},e_{4}}$,
$T_{e_{1},e_{2},e_{3},e_{4}}$ and $S_{e_{1},e_{2},e_{3},e_{4}}$ are
polynomials in the parameters $a_{i}$ of degrees $1-e_{1}-e_{2}-e_{3}-e_{4},$
$1-e_{1}-e_{2}-e_{3}-e_{4}$ and $2-e_{1}-e_{2}-e_{3}-e_{4}$, respectively,
while $c_{1}$, $c_{2}$ are of degree 0. If all parameters $a_{j}=0$ the
algebra is free. For these quantum quadratic algebras there is a natural
grading such that the operators $H$, $L_{j}$ are 2nd order and $X$ is 1st
order. The field of scalars can be either $\mathbb{R}$ or $\mathbb{C}$.

An abstract degenerate $($classical$)$ quadratic algebra ${\mathcal{Q}}$ is a
Poisson algebra with linearly independent generators ${\mathcal{X}}$,
${\mathcal{H}}$, ${\mathcal{L}}_{1}$, ${\mathcal{L}}_{2}$, and parameters
$a_{i}$, satisfying relations (1.1), (1.2), (1.3) with the commutator replaced
by the Poisson bracket, $H$, $L_{j}$, $X$ by ${\mathcal{H}}$,
${\mathcal{L}}_{j}$, ${\mathcal{X}}$, and the symmetrizer
$\\{L_{1}^{e_{1}},L_{2}^{e_{2}},X^{e_{3}}\\}$ by the product
${\mathcal{L}}_{1}^{e_{1}}{\mathcal{L}}_{2}^{e_{2}}{\mathcal{X}}^{e_{3}}/3!$.

These structures arise naturally in the study of classical and quantum
superintegrable systems in two dimensions, e.g., [23, 22], and, in the case of
zero potential systems, they are examples of Poisson structures, on which
there is a considerable literature [4, 6, 20]. A quantum 2D superintegrable
system is an integrable Hamiltonian system on a $2$-dimensional real or
complex Riemannian manifold with potential: $H=\Delta_{2}+V$, that admits $3$
algebraically independent partial differential operators commuting with $H$,
the maximum possible:

$\displaystyle[H,L_{j}]=0,\qquad L_{3}=H,\qquad j=1,2,3.$

Here $\Delta_{2}$ is the Laplace operator on the manifold. (We call this a
Helmholtz superintegrable system with eigenvalue equation $H\Psi=E\Psi$ to
distinguish it from a Laplace conformally superintegrable system,
$H\Psi=(\Delta_{2}+V)\Psi=0$ [18].) A system is of order $K$ if the maximum
order of the symmetry operators $L_{j}$ (other than $H$) is $K$; all such
systems are known for $K=1,2$ [3, 11, 14, 15]. Superintegrability captures the
properties of quantum Hamiltonian systems that allow the Schrödinger
eigenvalue problem $H\Psi=E\Psi$ to be solved exactly, analytically and
algebraically. A classical 2D superintegrable system is an integrable
Hamiltonian system on a real or complex $2$-dimensional Riemannian manifold
with potential: ${\mathcal{H}}=\sum\limits_{j,k=1}^{2}g^{jk}({\bf
x})p_{j}p_{k}+V({\bf x})$ in local coordinates $x_{1}$, $x_{2}$, $p_{1}$,
$p_{2}$ that admit 3 functionally independent phase space functions
${\mathcal{H}}$, ${\mathcal{L}}_{1}$, ${\mathcal{L}}_{2}$ in involution with
${\mathcal{H}}$, the maximum possible.

$\displaystyle\\{{\mathcal{H}},{\mathcal{L}}_{j}\\}=0,\qquad{\mathcal{L}}_{3}={\mathcal{H}},\qquad
j=1,2,3.$

A system is of order $K$ if the maximum order of the constants of the motion
$L_{j}$, $j\neq 3$, as polynomials in $p_{1}$, $p_{2}$ is $K$. Again all such
systems are known for $K=1,2$, and, for them, there is a 1-1 relationship
between classical and quantum 2nd order 2D superintegrable systems [13], i.e.,
the quantum system can be computed from the classical system, and vice versa.

The possible superintegrable systems divide into six classes:

  1. 1.

First order systems. These are the (zero-potential) Laplace–Beltrami
eigenvalue equations on constant curvature spaces. The symmetry algebras close
under commutation to form the Lie algebras ${\mathfrak{e}}(2,\mathbb{R})$,
${\mathfrak{e}}(1,1)$, ${\mathfrak{o}}(3,\mathbb{R})$ or
${\mathfrak{o}}(2,1)$. Such systems have been studied in detail, using group
theory methods.

  2. 2.

Free triplets. These are superintegrable systems with zero potential and all
generators of 2nd order. The possible spaces for which these systems can occur
were classified by Koenigs (1896). They are: constant curvature spaces, the
four Darboux spaces, and eleven 4-parameter Koenigs spaces [19]. In most cases
the symmetry operators will not generate a quadratic algebra, i.e., the
algebra will not close. If the system generates a nondegenerate quadratic
algebra we call it a free quadratic triplet.

  3. 3.

Nondegenerate systems. These are superintegrable systems with a non-zero
potential and the generating symmetries are all of 2nd order. The space of
potentials is 4-dimensional:

$\displaystyle V({\bf x})=a_{1}V_{(1)}({\bf x})+a_{2}V_{(2)}({\bf
x})+a_{3}V_{(3)}({\bf x})+a_{4}.$

The symmetry operators generate a nondegenerate quadratic algebra with
parameters $a_{j}$.

  4. 4.

Degenerate systems. There are 4 generators: one 1st order $X$ and 3 second
order $H$, $L_{1}$, $L_{2}$. Here, $X^{2}$ is not contained in the span of
$H$, $L_{1}$, $L_{2}$. The space of potentials is 2-dimensional: $V({\bf
x})=a_{1}V_{(1)}({\bf x})+a_{2}$. The symmetry operators generate a degenerate
quadratic algebra with parameters $a_{j}$. Relation (1.3) is an expression of
the fact that 4 symmetry operators cannot be algebraically independent. The
possible degenerate systems, classified up to conjugacy with respect to the
symmetry groups of their underlying spaces, are listed in Appendix A.

  5. 5.

Exceptional system. $E15$: $V=f(x-iy)$, $f$ an arbitrary function.

The exceptional case is characterized by the fact that the symmetry generators
are functionally linearly dependent [10, 12, 13, 15]. This is the only 2nd
order functionally linearly dependent 2D system but there are many such
systems in 3D, including the Calogero 3-body system on the line. In 3D such
systems have not yet been classified.

Every degenerate superintegrable system occurs as a restriction of the
3-parameter potentials to 1-parameter ones, such that one of the symmetries
becomes a perfect square: $L=X^{2}$. Here $X$ is a first order symmetry and a
new 2nd order symmetry appears so that this restriction admits more symmetries
than the original system, see Remark A.1. Basic results that relate these
superintegrable systems are the closure theorems:

######  Theorem 1.1.

A free triplet, classical or quantum, extends to a superintegrable system with
potential if and only if it generates a free quadratic algebra $\tilde{Q}$,
degenerate or nondegenerate.

######  Theorem 1.2.

A superintegrable system, degenerate or nondegenerate, classical or quantum,
with quadratic algebra $Q$, is uniquely determined by its free quadratic
algebra $\tilde{Q}$.

These theorems were proved for systems in [16]. The proofs are constructive:
Given a free quadratic algebra $\tilde{Q}$ one can compute the potential $V$
and the symmetries of the quadratic algebra $Q$. Thus as far as
superintegrable systems are concerned, all information about the systems is
contained in the free classical quadratic algebras.

######  Remark 1.3.

This paper is a companion to [5] where we studied nondegenerate quadratic
algebras, and we assume that the reader has some familiarity with this prior
work. In particular, Bôcher contractions, their properties and associated
notation, are treated there and we use them in this paper without detailed
comment.

The layout of this paper is as follows: In Section 2 we show how degenerate
Helmholtz superintegrable systems can be split into Stäckel equivalence
classes of Laplace conformally superintegrable systems and we determine how
each Helmholtz system can be characterized in its equivalence class. In
Section 3 we determine all Bôcher, i.e., geometrical, contractions of the
Laplace systems and obtain complete lists of the possible Helmholtz
contractions. In Section 4 we classify all abstract free quadratic algebras
and determine which of these can be realized as the quadratic algebra of a
Helmholtz degenerate superintegrable system. In Section 5 we classify all
abstract contractions of abstract free quadratic algebras and determine which
of these can be realized as Bôcher and Heisenberg contractions of the
quadratic algebras of Helmholtz degenerate superintegrable systems. In Fig. 2
we describe how restrictions of nondegenerate superintegrable systems to
degenerate ones, and contractions of degenerate superintegrable systems
account for the lower half of the Askey scheme. The upper half of the Askey
scheme is described by contractions of nondegenerate systems [5]. In Section 6
we assess our results. A list of all Helmholtz degenerate superintegrable
systems can be found in Appendix A.

##  2 Stäckel transforms and Laplace equations

Distinct degenerate classical or quantum superintegrable systems can be mapped
to one another by Stäckel transforms, invertible transforms that preserve the
structure of the quadratic algebra. This divides the 15 systems into 6 Stäckel
equivalence classes [22]. The most convenient way to understand the
equivalence classes is in terms of Laplace-like equations [18]. Since every 2D
space is conformally flat there always exist “Cartesian-like” coordinates $x$,
$y$ such that the Hamilton–Jacobi equation can be expressed in the form $H=E$
where $H=\frac{p_{x}^{2}+p_{y}^{2}}{\lambda(x,y)}+\alpha V$ and $\alpha$ is a
parameter. This is equivalent to the Laplace-like equation
$p_{x}^{2}+p_{y}^{2}+a_{1}V_{1}+a_{2}V_{2}=0$ where $V_{1}=\lambda V$,
$V_{2}=\lambda$, $a_{1}=\alpha$, $a_{2}=-E$, now with 2 parameters. Symmetries
(constants of the motion) for the Helmholtz equation correspond to conformal
symmetries of the Laplace equation. The Hamilton–Jacobi equation is defined on
one of a variety of conformally flat spaces but the Laplace equation is always
defined on flat space with conformal symmetry algebra
${\mathfrak{so}}(4,\mathbb{C})$ [18]. An important observation is that the
Laplace equations are Stäckel equivalence classes: two Helmholtz systems are
Stäckel equivalent if and only if they correspond to the same Laplace
equation.

######  Remark 2.1.

Indeed, If the Laplace conformally superintegrable equation can be split in
the form $p_{x}^{2}+p_{y}^{2}+V_{0}-{\tilde{E}}W=0$, where $\tilde{E}$ is an
arbitrary parameter, $W$ is a nonconstant function, and $V_{0}$, $W$ are
independent of $E$, then $W$, by division, defines a conformal Stäckel
transform to the superintegrable Helmholtz system
${\tilde{H}}=\frac{1}{W}(p_{x}^{2}+p_{y}^{2}+V_{0})={\tilde{E}}$. If the
Laplace system admits another splitting
$p_{x}^{2}+p_{y}^{2}+V^{\prime}_{0}-{\tilde{E}^{\prime}}W^{\prime}=0$, it
determines another superintegrable Helmholtz system
${\tilde{H}^{\prime}}=\frac{1}{W^{\prime}}(p_{x}^{2}+p_{y}^{2}+V^{\prime}_{0})={\tilde{E}^{\prime}}$
and $\tilde{H}^{\prime}$ can be obtained from ${\tilde{H}}$ by an invertible
Stäckel transform $\frac{W^{\prime}}{W}$. Thus all Helmholtz systems that can
be obtained from the Laplace equation by splitting the potential are Stäckel
equivalent to one another.

The Laplace equations for nondegenerate systems were derived in [18], see
Table 1. The Laplace equations for degenerate systems are listed in Table 2.
The notation $a_{i}$ in Table 2 describes how these systems can be obtained as
restrictions of systems in Table 1, but with added symmetry.

System | Non-degenerate potentials $V(x,y)$2pt2pt
---|---
$[1,1,1,1]$ |  $\frac{a_{1}}{x^{2}}+\frac{a_{2}}{y^{2}}+\frac{4a_{3}}{(x^{2}+y^{2}-1)^{2}}-\frac{4a_{4}}{(x^{2}+y^{2}+1)^{2}}$3pt3pt
$[2,1,1]$ |  $\frac{a_{1}}{x^{2}}+\frac{a_{2}}{y^{2}}-a_{3}(x^{2}+y^{2})+a_{4}$3pt
$[2,2]$ |  $\frac{a_{1}}{(x+iy)^{2}}+\frac{a_{2}(x-iy)}{(x+iy)^{3}}+a_{3}-a_{4}(x^{2}+y^{2})$3pt
$[3,1]$ |  $a_{1}-a_{2}x+a_{3}(4x^{2}+y^{2})+\frac{a_{4}}{y^{2}}$3pt
$[4]$ |  $a_{1}-a_{2}(x+iy)+a_{3}(3(x+iy)^{2}+2(x-iy))-a_{4}(4(x^{2}+y^{2})+2(x+iy)^{3})$3pt
$[0]$ |  $a_{1}-(a_{2}x+a_{3}y)+a_{4}(x^{2}+y^{2})$3pt
$(1)$ |  $\frac{a_{1}}{(x+iy)^{2}}+a_{2}-\frac{a_{3}}{(x+iy)^{3}}+\frac{a_{4}}{(x+iy)^{4}}$3pt
$(2)$ |  $a_{1}+a_{2}(x+iy)+a_{3}(x+iy)^{2}+a_{4}(x+iy)^{3}$3pt
Table 1: Four parameter Laplace systems. System | Degenerate potentials $V(x,y)$2pt2pt
---|---
$A$ |  $\frac{4\,a_{3}}{(x^{2}+y^{2}-1)^{2}}-\frac{4a_{4}}{(x^{2}+y^{2}+1)^{2}}$2pt2pt
$B$ |  $\frac{a_{1}}{x^{2}}+a_{4}$2pt
$C$ |  $a_{3}-a_{4}(x^{2}+y^{2})$2pt
$D$ |  $a_{1}-a_{2}x$2pt
$E$ |  $\frac{a_{1}}{(x+iy)^{2}}+a_{3}$2pt
$F$ |  $a_{1}-a_{2}(x+iy)$2pt
Table 2: Two-parameter Laplace systems.

The Helmholtz systems corresponding to each Laplace system are:

Stäckel equivalence classes: Here the notation refers to the Helmholtz
superintegrable systems listed in the Appendix.

  1. 1.

Class $A$ $(a_{3},a_{4})$: System $S3$ corresponds to $(1,0)$ and $(0,1)$.
System $S6$ corresponds to $(1,1)$. System $D4(b)D$ corresponds to
$(a_{3},a_{4})$ with $a_{3}a_{4}(a_{3}-a_{4})\neq 0$.

  2. 2.

Class $B$ $(a_{1},a_{4})$: System $S5$ corresponds to $(1,0)$. System $E6$
corresponds to $(0,1)$. System $D2D$ corresponds to $(a_{1},a_{4})$ with
$a_{1}a_{4}\neq 0$.

  3. 3.

Class $C$ $(a_{3},a_{4})$: System $E3$ corresponds to $(1,0)$. System $E{18}$
corresponds to $(0,1)$. System $D3E$ corresponds to $(a_{3},a_{4})$ with
$a_{3}a_{4}\neq 0$.

  4. 4.

Class $D$ $(a_{1},a_{2})$: System $E5$ corresponds to $(1,0)$. System $D1D$
corresponds to $(a_{1},a_{2})$ with $a_{1}a_{2}\neq 0$.

  5. 5.

Class $E$ $(a_{1},a_{3})$: System $E{14}$ corresponds to $(0,1)$. System
$E{12}$ corresponds to $(a_{1},a_{3})$ with $a_{1}a_{3}\neq 0$.

  6. 6.

Class $F$ $(a_{1},a_{2})$: System $E{13}$ corresponds to $(a_{1},a_{2})$ with
$a_{2}\neq 0$. System $E{4}$ corresponds to $(1,0)$.

Here, for example, system $D3E$ belongs to class $C$ and is obtained from the
Laplace equation by dividing it by $a_{3}-a_{4}(x^{2}+y^{2})$ where
$a_{3}a_{4}\neq 0$, whereas $E{18}$ is obtained by the same division with
$a_{3}=0$, $a_{4}=1$.

The conformal symmetry of these Laplace equations is best exploited by using
tetraspherical coordinates to linearize the action of the conformal symmetry
group [1, 18]. These are projective coordinates $x_{1}$, $x_{2}$, $x_{3}$,
$x_{4}$ on the null cone $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=0$, related
to flat space coordinates $x$, $y$ by

$\displaystyle x_{1}=2XT,\qquad x_{2}=2YT,\qquad
x_{3}=X^{2}+Y^{2}-T^{2},\qquad x_{4}=i\big{(}X^{2}+Y^{2}+T^{2}\big{)},$
$\displaystyle x=\frac{X}{T}=-\frac{x_{1}}{x_{3}+ix_{4}},\qquad
y=\frac{Y}{T}=-\frac{x_{2}}{x_{3}+ix_{4}},\qquad
x=\frac{s_{1}}{1+s_{3}},\qquad y=\frac{s_{2}}{1+s_{3}},$
$\displaystyle{\mathcal{H}}\equiv
p_{x}^{2}+p_{y}^{2}+{V}=(x_{3}+ix_{4})^{2}\left(\sum_{k=1}^{4}p_{x_{k}}^{2}+V_{B}\right)=(1+s_{3})^{2}\left(\sum_{j=1}^{3}p_{s_{j}}^{2}+V_{S}\right),$
$\displaystyle
V=(x_{3}+ix_{4})^{2}V_{B},\\!\\!\\!\qquad(1+s_{3})=-i\frac{(x_{3}+ix_{4})}{x_{4}},\\!\\!\\!\qquad
s_{1}=\frac{ix_{1}}{x_{4}},\\!\\!\\!\qquad
s_{2}=\frac{ix_{2}}{x_{4}},\\!\\!\\!\qquad s_{3}=\frac{-ix_{3}}{x_{4}}.$

Thus the Laplace equation ${\mathcal{H}}\equiv p_{x}^{2}+p_{y}^{2}+{V}=0$ in
Cartesian coordinates becomes $\sum\limits_{k=1}^{4}p_{x_{k}}^{2}+V_{B}=0$ in
tetraspherical coordinates. Here, the $s_{j}$ refer to coordinates on the unit
2-sphere: $s_{1}^{2}+s_{2}^{2}+s_{3}^{2}=1$,

The possible limits of one superintegrable system to another can be derived
and classified by using tetraspherical coordinates and special Bôcher
contractions of ${\mathfrak{so}}(4,\mathbb{C})$ to itself. The method is
described in detail in [5, 18]. Here we just recall the basic definition of a
Bôcher contraction, Let ${\bf x}={\bf A}(\epsilon){\bf y}$, and ${\bf
x}=(x_{1},\dots,x_{4})$, ${\bf y}=(y_{1},\dots,y_{4})$ be column vectors, and
${\bf A}=(A_{jk}(\epsilon))$ be a $4\times 4$ matrix with matrix elements

$\displaystyle
A_{kj}(\epsilon)=\sum_{\ell=-N}^{N}a^{\ell}_{kj}\epsilon^{\ell},$

where $N$ is a nonnegative integer and the $a^{\ell}_{kj}$ are complex
constants. (Here, $N$ can be arbitrarily large, but it must be finite in any
particular case.) We say that the matrix $\bf A$ defines a Bôcher contraction
of the conformal algebra ${\mathfrak{so}}(4,\mathbb{C})$ to itself provided

$\displaystyle 1)\quad\det({\bf A})=\pm 1,\ {\rm constant\ for\ all\
}\epsilon\neq 0,$ $\displaystyle 2)\quad{\bf x}\cdot{\bf
x}\equiv\sum_{j=1}^{4}x_{i}(\epsilon)^{2}={\bf y}\cdot{\bf y}+O(\epsilon).$

If, in addition, ${\bf A}\in O(4,\mathbb{C})$ for all $\epsilon\neq 0$ the
matrix $\bf A$ defines a special Böcher contraction. For a special Böcher
contraction ${\bf x}\cdot{\bf x}={\bf y}\cdot{\bf y}$, with no error term.
(These contractions correspond to limit relations introduced by Bôcher to
obtain all orthogonal separable coordinates for Laplace and wave equations as
limits of cyclidic coordinates. There is an infinite family of such
contractions, but they can be generated by 4 basic contractions.) Related
contraction methods that don’t make use of tetraspherical coordinates directly
can be found in references such as [8, 9].

Bôcher contractions take a Laplace system to itself. The contraction process
has already been described in [5, 18] and references therein, but we discuss,
briefly, the main ideas. Suppose we have a degenerate Laplace superintegrable
system with potential $V({\bf x},{\bf a})=a_{1}V_{1}({\bf x})+a_{2}V_{2}({\bf
x})$ and generating conformal symmetries $X={\cal X}+W_{0}$, $L_{1}={\cal
L}_{1}+W_{1}$, $L_{2}={\cal L}_{2}+W_{2}$, where ${\cal X}$, ${\cal L}_{1}$,
${\cal L}_{2}$ are free 2nd order conformal symmetries and $W_{0}$, $W_{1}$,
$W_{2}$ are functions of the tetraspherical coordinates $x_{i}$. Applying a
Bôcher contraction $A(\epsilon)$ to the free symmetries we obtain

$\displaystyle{\cal X}(\epsilon)=\epsilon^{\alpha_{0}}{\cal
X}^{\prime}+O(\epsilon),\qquad{\cal
L}_{j}(\epsilon)=\epsilon^{\alpha_{j}}{\cal L}_{j}^{\prime}+O(\epsilon),\qquad
j=1,2,$

where ${\cal X}^{\prime}\in{\mathfrak{so}}(4,{\mathbb{C}})$ and the ${\cal
L}_{j}^{\prime}$ are quadratic in ${\mathfrak{so}}(4,{\mathbb{C}})$. By a
change of basis $\\{L_{1},L_{2}\\}$ if necessary, one can verify that ${\cal
X}^{\prime}$, ${\cal L}_{j}^{\prime}$, $j=1,2$ generate a free conformal
quadratic algebra, The action of the Bôcher contraction on the 2-dimensional
potential space preserves its dimension and maps it smoothly as a function of
$\epsilon$, as follows from an examination of the Bertrand-Darboux equations.
Thus we get a 2-dimensional potential space in the limit. To find an explicit
basis for the contracted potential $V^{\prime}({\bf b},{\bf
y})=b_{1}V_{1}^{\prime}({\bf y})^{\prime}+b_{2}V_{2}({\bf y})^{\prime}$ we put
$a_{1}=\sum\limits_{k=-\infty}^{\infty}c_{k}\epsilon^{k}$,
$a_{2}=\sum\limits_{k=-\infty}^{\infty}d_{k}\epsilon^{k}$, where only a finite
number of the coefficients $c_{k}$, $d_{k}$ can be nonzero. Then it is a
linear algebra problem to determine the $c_{k}$, $d_{k}$ such that
$\lim\limits_{\epsilon\to 0}V({\bf x}(\epsilon),{\bf
a}(\epsilon))=V^{\prime}({\bf b},{\bf y})$ exists for independent potential
functions $V_{1}^{\prime}({\bf y})^{\prime}$, $V_{2}^{\prime}({\bf
y})^{\prime}$, where the nonzero $c_{k}$, $d_{k}$ are linear in $b_{1}$,
$b_{2}$. The limit is guaranteed to exist and is unique up to a change of
basis $\\{V_{1}^{\prime}({\bf y})^{\prime},V_{2}^{\prime}({\bf
y})^{\prime}\\}$ for the target potential. Only the last limit and the linear
algebra problem need to be solved to identify the contraction. This work was
carried out with the assistance of the symbol manipulation programs Maple and
Mathematica. There is one additional complication; the results of the
contraction are not invariant under a permutation of the indices of the
hyperspherical coordinates defining the contraction matrix $A_{ij}$. Thus one
Bôcher contraction applied to a source system can yield a multiplicity of
target results, and all permutations need to be examined.

The results are rather complicated. Fig. 1 provides a clearer idea of what is
happening. There are 4 basic Böcher contractions of 2D Laplace systems and
each one when applied to a Laplace system, and each permutation treated,
yields another Laplace superintegrable system. A system in class $K_{1}$ can
be obtained from a system in class $K_{2}$ via contraction provided there is a
directed arrow path from $K_{2}$ to $K_{1}$. All systems follow from $A$ for
degenerate potentials, and $A$ is a restriction of $[1111]$ with increased
symmetry. Fig. 1 describes only the existence or nonexistence of contractions,
not the multiplicity of distinct contractions.

Figure 1: Contraction relations for degenerate Laplace systems.

Our basic interest, however, is in Helmholtz contractions, i.e., contractions
of a Helmholtz superintegrable system to another such system. The key is to
start with a Laplace system, take a conformal Stäckel transform to a Helmholtz
system (which we initially interpret as another Laplace system) and then take
a Bôcher contraction of the new system, which as described below gives a new
Helmholtz system. The result is the contraction of one Helmholtz system to
another This can be done in such a way the “diagrams commute”, i.e., a
Helmholtz contraction is induced by a Bôcher contraction and a Stäckel
transform [18]. For example, let $\cal H$ be the Hamiltonian for class $A$. In
terms of tetraspherical coordinates a general conformal Stäckel transformed
potential will take the form

$\displaystyle
V=\frac{\frac{a_{3}}{x_{3}^{2}}+\frac{a_{4}}{x_{4}^{2}}}{\frac{b_{3}}{x_{3}^{2}}+\frac{b_{4}}{x_{4}^{2}}}=\frac{V_{A}}{F({\bf
x},{\bf b})},$

where $F({\bf x},{\bf b})=\frac{b_{3}}{x_{3}^{2}}+\frac{b_{4}}{x_{4}^{2}}$,
and the transformed Hamiltonian will be ${\hat{\cal H}}=\frac{1}{F({\bf
x},{\bf b})}{\cal H}$, where the transform is determined by the fixed vector
$(b_{3},b_{4})$. Now we apply the Bôcher contraction $[1,1,1,1]\to[2,1,1]$ to
this system. Depending on the permutation of the indices $x_{j}$, in the limit
as $\epsilon\to 0$ the potential $V_{A}\to V_{B}$, or $V_{A}\to V_{C}$, and
$\cal H\to{\cal H}^{\prime}$, the $B$ or $C$ system. Now consider $F({\bf
x}(\epsilon),{\bf b})=V^{\prime}({\bf
x}^{\prime},b)\epsilon^{\alpha}+O\big{(}\epsilon^{\alpha+1}\big{)}$, where the
integer exponent $\alpha$ depends upon our choice of $\bf b$. From our theory,
the system defined by Hamiltonian ${\hat{\cal
H}}^{\prime}=\lim\limits_{\epsilon\to 0}\epsilon^{\alpha}{\hat{\cal
H}}(\epsilon)=\frac{1}{V^{\prime}({\bf x}^{\prime},{\bf b})}{\cal H}^{\prime}$
is a superintegrable system that arises from the system $A$ by a conformal
Stäckel transform induced by the potential $V^{\prime}({\bf x}^{\prime},{\bf
b})$. Thus the Helmholtz superintegrable system with potential $V=V_{A}/F$
contracts to the Helmholtz superintegrable system with potential
$V_{S}/V^{\prime}$, where $S=B$ or $S=C$. The contraction is induced by a
generalized Inönü–Wigner Lie algebra contraction of the conformal algebra
${\mathfrak{so}}(4,\mathbb{C})$. Always the $V^{\prime}$ can be identified
with a specialization of the $S$ potential. Thus a conformal Stäckel transform
of $A$ has been contracted to a conformal Stäckel transform of $S$.

##  3 Degenerate Helmholtz contractions

The superscript for each targeted Helmholtz system is the value of the
exponent $\alpha$ associated with the contraction. In each table below,
corresponding to a single Laplace equation equivalence class, the top line is
a list of the Helmholtz systems in the class, and the lower lines are the
target systems under the Bôcher contraction.

$\displaystyle\begin{array}[]{llllllll}&A&{\rm equivalence}&{\rm class}&{\rm
contractions}&&\\\ \hline\cr{\rm contraction}&{S3}&S6&D4D\\\
\hline\cr{[1111]}\downarrow[211]&E3^{2}&E{18}^{4}&E3^{2}\tsep{2pt}\\\
&S5^{0},\ E6^{2}&S5^{0}&S5^{0}\\\
\hline\cr{[1111]}\downarrow[22]&E3^{2}&E{18}^{4}&E{3}^{2}\tsep{2pt}\\\
&E{14}^{2}&E{12}^{2}&E{12}^{2}\\\
\hline\cr{[1111]}\downarrow[31]&E5^{2}&E5^{2}&E5^{2},\ D1D^{3}\tsep{2pt}\\\
&E6^{2},\ S5^{0}&S5^{0}&S5^{0}\\\
\hline\cr{[1111]}\downarrow[4]&E{4}^{6}&E{4}^{6}&E{4}^{6},\
E{13}^{8}\tsep{2pt}\\\ \hline\cr\end{array}$
$\displaystyle\begin{array}[]{llllllll}&B&{\rm equivalence}&{\rm class}&{\rm
contractions}&&\\\ \hline\cr{\rm contraction}&{S5}&E6&D2D\\\
\hline\cr{[1111]}\downarrow[211]&E{14}^{2}&E{14}^{0}&E{14}^{0}\tsep{2pt}\\\
&S5^{0}&E6^{2}&S5^{0}\\\
\hline\cr{[1111]}\downarrow[22]&E{14}^{2}&E{14}^{2}&E{12}^{2}\tsep{2pt}\\\
\hline\cr{[1111]}\downarrow[31]&E5^{2}&E5^{2}&E5^{2},\ D1D^{3}\tsep{2pt}\\\
&S5^{0}&E6^{2}&S5^{0},\ E6^{2}\\\
\hline\cr{[1111]}\downarrow[4]&E{4}^{6}&E{4}^{6}&E{4}^{6},\
E{13}^{8}\tsep{2pt}\\\ \hline\cr\end{array}$
$\displaystyle\begin{array}[]{llllllll}&C&{\rm equivalence}&{\rm class}&{\rm
contractions}&&\\\ \hline\cr{\rm contraction}&{E3}&E{18}&D3E\tsep{2pt}\\\
\hline\cr{[1111]}\downarrow[211]&E{5}^{2}&E{5}^{2}&E{5}^{2,3}\tsep{2pt}\\\
&E3^{2}&E{18}^{4}&E3^{2}\\\
\hline\cr{[1111]}\downarrow[22]&E3^{2}&E{18}^{2,4}&E3^{2},\
D3E^{2}\tsep{2pt}\\\ \hline\cr[1111]\downarrow[31]&E5^{2}&E5^{2}&E5^{2},\
D1D^{3}\tsep{2pt}\\\
\hline\cr{[1111]}\downarrow[4]&E{4}^{6}&E{4}^{6}&E{4}^{6},\
E{13}^{8}\tsep{2pt}\\\ \hline\cr\end{array}$
$\displaystyle\begin{array}[]{llllllll}&D&{\rm equivalence}&{\rm class}&{\rm
contractions}&&\\\ \hline\cr{\rm contraction}&{E5}&D1D\\\
\hline\cr{[1111]}\downarrow[211]&E{4}^{0}&E{13}^{-1},\ E4^{0}\tsep{2pt}\\\
&E5^{2}&E5^{2},\ D1D^{3}\\\ \hline\cr{[1111]}\downarrow[22]&E4^{2}&E{4}^{2},\
E{13}^{2}\tsep{2pt}\\\ \hline\cr{[1111]}\downarrow[31]&E5^{2}&E5^{2},\
D1D^{2}\tsep{2pt}\\\ \hline\cr[1111]\downarrow[4]&E{4}^{6}&E{4}^{6},\
E{13}^{8}\tsep{2pt}\\\ \hline\cr\end{array}$
$\displaystyle\begin{array}[]{llllllll}&E&{\rm equivalence}&{\rm class}&{\rm
contractions}&&\\\ \hline\cr{\rm contraction}&E{14}&E{12}\\\
\hline\cr{[1111]}\downarrow[211]&E{4}^{2}&E{4}^{2},\ E{13}^{3}\tsep{2pt}\\\
&E{14}^{0}&E{14}^{0}\\\ \hline\cr{[1111]}\downarrow[22]&E4^{2}&E{4}^{2},\
E{13}^{4}\tsep{2pt}\\\ &E{14}^{2}&E{12}^{2}\\\
\hline\cr{[1111]}\downarrow[31]&E4^{2}&E4^{2},\ E{13}^{3}\tsep{2pt}\\\
\hline\cr[1111]\downarrow[4]&E{4}^{6}&E{4}^{6},\ E{13}^{8}\tsep{2pt}\\\
\hline\cr\end{array}$ $\displaystyle\begin{array}[]{llllllll}&F&{\rm
equivalence}&{\rm class}&{\rm contractions}&&\\\ \hline\cr{\rm
contraction}&{E4}&E{13}\\\
\hline\cr{[1111]}\downarrow[211]&E{4}^{0,2}&E{13}^{-1,3},\ E4^{2}\tsep{2pt}\\\
\hline\cr{[1111]}\downarrow[22]&E4^{2}&E{13}^{2}\tsep{2pt}\\\
\hline\cr{[1111]}\downarrow[31]&E4^{2}&E4^{2}\tsep{2pt}\\\
\hline\cr{[1111]}\downarrow[4]&E{4}^{6}&E{4}^{6}\tsep{2pt}\\\
\hline\cr\end{array}$

##  4 Classification of free abstract degenerate
classical quadratic algebras

###  4.1 Abstract quadratic algebras

The special case of a 2D degenerate classical superintegrable system on a
constant curvature space or a Darboux space with all parameters equal to zero
(no potential) gives rise to a special kind of quadratic algebra which we call
a free abstract quadratic algebra. Below is a precise definition.

######  Definition 4.1.

An abstract 2D free degenerate (classical) quadratic algebra is a complex
Poisson algebra possessing a linearly independent generating set
$\\{\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3}=\mathcal{H},\mathcal{X}\\}$
that satisfy:

  1. 1.

The associative product is abelian.

  2. 2.

$\mathcal{A}$ is graded: $\mathcal{A}=\oplus_{k=0}^{\infty}\mathcal{A}_{k}$
where each $\mathcal{A}_{k}$ is a complex vector space, the associative
product takes $\mathcal{A}_{k}\times\mathcal{A}_{l}$ into $\mathcal{A}_{k+l}$,
and the Poisson brackets $\\{\\_\\_,\\_\\_\\}$ goes from
$\mathcal{A}_{k}\times\mathcal{A}_{l}$ into $\mathcal{A}_{k+l-1}$.

  3. 3.

$\mathcal{A}_{0}=\mathbb{C}$.

  4. 4.

$\mathcal{A}_{1}=\mathbb{C}\mathcal{X}$.

  5. 5.

$\mathcal{A}_{2}=\operatorname{span}\\{\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{X}^{2}\\}$.

  6. 6.

The elements
$\\{\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4}:=\mathcal{X}^{2}\\}$
satisfy a relation given by a homogeneous polynomial of degree 2,
${G}(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4})=0$.

  7. 7.

Any nonzero polynomial $F$ of minimal degree such that
${F}(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4})=0$ is a
multiple of $G$.

  8. 8.

The center of $\mathcal{A}$ contains no elements of order $1$ and any element
of order two in the center is a multiple of $\mathcal{L}_{3}=\mathcal{H}$.

  9. 9.

The polynomial $G$ depends non-trivially on at least one of the non-central
2nd order generators ${\mathcal{L}}_{1}$, ${\mathcal{L}}_{2}$.

We shall simply refer to such an algebra as a free abstract quadratic algebra.

######  Remark 4.2.

A free abstract quadratic algebra is a special case of a three-dimensional
affine Poisson variety, see, e.g., [4, 20]. Description of some of the
properties of such Poisson varieties, along with classification of such
Poisson structures on the affine space $\mathbb{C}^{3}$ can be found in [20,
Section 9.2]. Poisson structures on the manifold $\mathbb{R}^{3}$ were
classified at [6]. It should be noted that the Poisson algebras considered in
this paper, are not quadratic Poisson structures in the sense of [20, Section
8.2]. In addition our classification scheme is different from the above
mentioned classifications.

######  Theorem 4.3.

Keeping the same notations as of Definition 4.1, let $\mathcal{A}$ be a free
abstract quadratic algebra. Then there exists a non zero $K\in\mathbb{C}$ such
that:

$\displaystyle\\{\mathcal{L}_{1},\mathcal{L}_{2}\\}=K\frac{\partial
G}{\partial{\cal X}},\qquad\\{{\cal X},\mathcal{L}_{1}\\}=K\frac{\partial
G}{\partial\mathcal{L}_{2}},\qquad\\{{\cal
X},\mathcal{L}_{2}\\}=-K\frac{\partial G}{\partial\mathcal{L}_{1}}.$ (4.1)

$($We shall refer to equations (4.1) as the structure equations of
$\mathcal{A})$.

###### Proof.

Note that
${G}(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4})=0$
implies that

$\displaystyle 0=\\{\mathcal{X},G\\}=\frac{\partial
G}{\partial\mathcal{L}_{1}}\\{{\cal X},\mathcal{L}_{1}\\}+\frac{\partial
G}{\partial\mathcal{L}_{2}}\\{{\cal X},\mathcal{L}_{2}\\}.$

The above equation is a polynomial expression in $\chi^{2},\mathcal{L}_{1}$,
$\mathcal{L}_{2}$, $\mathcal{H}$. Any one of the terms $\frac{\partial
G}{\partial\mathcal{L}_{1}}$, $\\{{\cal X},\mathcal{L}_{1}\\}$,
$\frac{\partial G}{\partial\mathcal{L}_{2}}$, $\\{{\cal X},\mathcal{L}_{2}\\}$
is either zero or a polynomial of degree one in the variables $\chi^{2}$,
$\mathcal{L}_{1}$, $\mathcal{L}_{2}$, $\mathcal{H}$. Since at least one of the
terms $\frac{\partial G}{\partial\mathcal{L}_{1}}$, $\frac{\partial
G}{\partial\mathcal{L}_{2}}$ is non-zero then we must have

$\displaystyle\\{{\cal X},\mathcal{L}_{1}\\}=K\frac{\partial
G}{\partial\mathcal{L}_{2}},\qquad\\{{\cal
X},\mathcal{L}_{2}\\}=-K\frac{\partial G}{\partial\mathcal{L}_{1}},$

for some $K$. If $K=0$ then the first order element ${\cal X}$ must be in the
center of $\mathcal{A}$ which is impossible. Hence $K\neq 0$. Similarly
${G}(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4})=0$
implies

$\displaystyle 0=\\{\mathcal{L}_{1},G\\}=\frac{\partial
G}{\partial\mathcal{X}}\\{\mathcal{L}_{1},{\cal X}\\}+\frac{\partial
G}{\partial\mathcal{L}_{2}}\\{\mathcal{L}_{1},\mathcal{L}_{2}\\},$
$\displaystyle 0=\\{\mathcal{L}_{2},G\\}=\frac{\partial
G}{\partial\mathcal{X}}\\{\mathcal{L}_{2},{\cal X}\\}+\frac{\partial
G}{\partial\mathcal{L}_{1}}\\{\mathcal{L}_{2},\mathcal{L}_{1}\\},$

and along with our assumption that at least one of the terms $\frac{\partial
G}{\partial\mathcal{L}_{1}}$, $\frac{\partial G}{\partial\mathcal{L}_{2}}$ is
non-zero we see that $\\{\mathcal{L}_{1},\mathcal{L}_{2}\\}=K\frac{\partial
G}{\partial{\cal X}}$. ∎

######  Corollary 4.4.

Keeping the same notations as of Definition 4.1, let $\mathcal{A}$ be a free
abstract quadratic algebra. Then
${G}(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4})$ depends
non-trivially in at least two of the three generators $\mathcal{L}_{1}$,
$\mathcal{L}_{2}$, $\mathcal{L}_{4}$.

###### Proof.

It easily follows from the structure equations (4.1) that if
${G}(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4})$ depends
on at most one of the generators $\mathcal{L}_{1}$, $\mathcal{L}_{2}$,
$\mathcal{L}_{4}$ then the center of $\mathcal{A}$ contains a second order
generator which is not a multiple of $\mathcal{H}$. ∎

Free abstract quadratic algebras appear as symmetry algebras of 2D degenerate
free superintegrable systems. That is the main motivation for their study. For
a given $G$ there is a 1-parameter family of quadratic algebras, parametrized
by $K$. However, these algebras are isomorphic. It follows that the particular
choice of nonzero $K$ in the classification theory to follow is immaterial.

######  Note 4.5.

If phase space generators ${\mathcal{L}}_{1}$, ${\mathcal{L}}_{2}$,
${\mathcal{H}}$, ${\mathcal{X}}$ satisfy a 4th order (in the momentum
variables) relation ${G}=0$ but the closure relations for
$\\{{\mathcal{X}},{\mathcal{L}}_{1}\\}$,
$\\{{\mathcal{X}},{\mathcal{L}}_{2}\\}$,
$\\{{\mathcal{L}}_{1},{\mathcal{L}}_{2}\\}$, are not satisfied, then $K$ is
rational in the generators.

For quantum degenerate systems, knowledge of the Casimir relation
$G(L_{1},L_{2},H,X,\alpha)=0$ is sufficient to determine the quadratic
algebra, subject to the same conditions as the classical case. Details can be
found in [7].

###  4.2 The symmetry group of a free abstract quadratic algebra

In this section we determine the symmetry group of a free abstract quadratic
algebra. Note that once we fixed a generating set
$\\{\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},{\cal X}\\}$, the Casimir
$G$ is given by a symmetric $4\times 4$ matrix $B(G)$ defined by

$\displaystyle{G}(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4})=\sum_{i,j=1}^{4}B({G})_{ij}\mathcal{L}_{i}\mathcal{L}_{j},$

The pair $(B(G),K)$ (consisting of the Casimir and the multiplicative constant
in the structure equations) is defined only up to a constant. That is, the
pair $\big{(}zB(G),z^{-1}K\big{)}$ for any nonzero $z$ can also serve as a
Casimir and a structure constant for the same quadratic algebra and the same
basis. A free abstract quadratic algebra is completely determined by a
generating set $\\{\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},{\cal
X}\\}$ (satisfying the conditions of Definition 4.1) together with the pair
$(B(G),K)$. As noted previously, the constant $K$ can always be normalized to
$1$. Given two sets of generators
$\\{\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},{\cal X}\\}$,
$\big{\\{}\widetilde{\mathcal{L}}_{1},\widetilde{\mathcal{L}}_{2},\widetilde{\mathcal{L}}_{3},\widetilde{{\cal
X}}\big{\\}}$ of the same algebra $\mathcal{A}$ such that:

  1. 1.

$2=\operatorname{deg}(\mathcal{L}_{1})=\operatorname{deg}\big{(}\widetilde{\mathcal{L}}_{1}\big{)}=\operatorname{deg}(\mathcal{L}_{2})=\operatorname{deg}(\widetilde{\mathcal{L}}_{2})=\operatorname{deg}(\mathcal{L}_{3})=\operatorname{deg}\big{(}\widetilde{\mathcal{L}}_{3}\big{)}$.

  2. 2.

$1=\operatorname{deg}(\mathcal{X})=\operatorname{deg}\big{(}\widetilde{\mathcal{X}}\big{)}$.

  3. 3.

${\mathcal{L}}_{3}$ and $\widetilde{\mathcal{L}}_{3}$ are in the center of
$\mathcal{A}$.

Then there is a ’change of basis matrix’ $A$ of the form

$\displaystyle A:=\left(\begin{matrix}A_{1,1}&A_{1,2}&A_{1,3}&A_{1,4}&0\\\
A_{2,1}&A_{2,2}&A_{2,3}&A_{2,4}&0\\\ 0&0&A_{3,3}&0&0\\\ 0&0&0&A_{4,4}&0\\\
0&0&0&0&A_{5,5}\end{matrix}\right)\in{\rm GL}(5,\mathbb{C}),$ (4.2)

with $A_{4,4}=A_{5,5}^{2}$ such that for
$\mathcal{L}:=(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4},\mathcal{X})^{\rm
t}$,
$\widetilde{\mathcal{L}}:=\big{(}\widetilde{\mathcal{L}}_{1},\widetilde{\mathcal{L}}_{2},\widetilde{\mathcal{L}}_{3},\widetilde{\mathcal{L}}_{4},\widetilde{\mathcal{X}}\big{)}^{\rm
t}$

$\displaystyle{\mathcal{L}}=A\widetilde{\mathcal{L}}.$ (4.3)

For $A$ as above we define

$\displaystyle\widehat{A}=\left(\begin{matrix}A_{1,1}&A_{1,2}&A_{1,3}&A_{1,4}\\\
A_{2,1}&A_{2,2}&A_{2,3}&A_{2,4}\\\ 0&0&A_{3,3}&0\\\
0&0&0&A_{4,4}\end{matrix}\right).$

Besides linear change of basis we can also rescale the invariants $(B(G),K)$.
The full symmetry group of a free abstract quadratic algebra is given by the
collection of pairs $(A,z)$, $A$ as in equation (4.2) and
$z\in\mathbb{C}^{*}$. We shall denote this group by $G_{\operatorname{degn}}$.
The action of such $(A,z)\in G_{\operatorname{degn}}$ on the generators is
given by equation (4.3) and on $(G,K)$ by

$\displaystyle B(G^{\prime})=z\widehat{A}^{\rm t}B({G})\widehat{A},$
$\displaystyle
K^{\prime}=z^{-1}K(A_{1,1}A_{2,2}-A_{1,2}A_{2,1})^{-1}A_{5,5}^{-1},$

where $B({G})$ is the symmetric matrix representing $G$ with respect to
$\\{\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4}\\}$, and
$B(G^{\prime})$ is the symmetric matrix representing $G^{\prime}:=(A,z)\cdot
G$ with respect to
$\big{\\{}\widetilde{\mathcal{L}}_{1},\widetilde{\mathcal{L}}_{2},\widetilde{\mathcal{L}}_{3},\widetilde{\mathcal{L}}_{4}\big{\\}}$.
We shall call the subgroup of $G_{\operatorname{degn}}$ of all pairs of the
form $(A,1)$ as the group of linear change of bases,we denote it by
$G_{\operatorname{degn}}^{\operatorname{lin}}$. In the following we shall
determine a canonical form for the pair $(B(G),K)$ of each abstract quadratic
algebra. This means that we shall decide upon a unique representative from
each orbit of $G_{\operatorname{degn}}$ in the space of all possible pairs
$(B(G),K)$ that arise from an abstract quadratic algebra. Without loss of
generality we can assume that $K=1$ and determine representative from each
orbit of the subgroup of $G_{\operatorname{degn}}$ that fixes the value of
$K$. Explicitly this group is given by

$\displaystyle G_{\operatorname{degn}}^{K}:=\big{\\{}(A,z)\in
G_{\operatorname{degn}}\,|\,z=(A_{1,1}A_{2,2}-A_{1,2}A_{2,1})^{-1}A_{5,5}^{-1}\big{\\}}.$

###  4.3 The canonical form

In this section after we identify some invariants of orbits of $(B(G),K)$
under the action of $G_{\operatorname{degn}}^{\operatorname{K}}$ we consider
the reduction to a canonical form. That is, for each orbit we choose a
representative which we call the canonical form $(B(G),K)$. For a given
Casimir $(B(G),K)$ of a free abstract quadratic algebra $\mathcal{A}$ and a
given $(A,z)\in G_{\operatorname{degn}}^{\operatorname{K}}$ we introduce the
following notations:

$\displaystyle\widehat{A}=\left(\begin{matrix}A_{1,1}&A_{1,2}&A_{1,3}&A_{1,4}\\\
A_{2,1}&A_{2,2}&A_{2,3}&A_{2,4}\\\ 0&0&A_{3,3}&0\\\
0&0&0&A_{4,4}\end{matrix}\right)=\left(\begin{matrix}r&s\\\
0&t\end{matrix}\right),$ $\displaystyle
B({G})=\left(\begin{matrix}b_{1,1}&b_{1,2}&b_{1,3}&b_{1,4}\\\
b_{1,2}&b_{2,2}&b_{2,3}&b_{2,4}\\\ b_{1,3}&b_{2,3}&b_{3,3}&b_{3,4}\\\
b_{1,4}&b_{2,4}&b_{3,4}&b_{4,4}\end{matrix}\right)=\left(\begin{matrix}b&c\\\
c^{\rm t}&d\end{matrix}\right).$

Direct calculation shows that

$\displaystyle\widehat{A}^{\rm t}B({G})\widehat{A}=\left(\begin{matrix}r^{\rm
t}br&r^{\rm t}(bs+ct)\\\ s^{\rm t}br+tc^{\rm t}r&s^{\rm t}(bs+ct)+t(c^{\rm
t}s+dt)\end{matrix}\right).$

Hence under the action $G_{\operatorname{degn}}^{\operatorname{K}}$ the rank
of $B(G)$ and the rank of its upper left 2 by 2 block (which we denote by $b$)
are preserved. We use the theory of symmetric bilinear forms and find
$(A,z)\in G_{\operatorname{degn}}^{\operatorname{K}}$ that cast $B(G)$ into
exactly one of the following forms depending on $\operatorname{rank}(b)$.

$\displaystyle 1.\quad B(G)=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&b_{3,3}&b_{3,4}\\\
0&0&b_{3,4}&b_{4,4}\end{matrix}\right)\qquad\text{if}\quad\operatorname{rank}(b)=2,$
$\displaystyle 2.\quad B(G)=\left(\begin{matrix}1&0&0&0\\\
0&0&b_{2,3}&b_{2,4}\\\ 0&b_{2,3}&b_{3,3}&b_{3,4}\\\
0&b_{2,4}&b_{3,4}&b_{4,4}\end{matrix}\right)\qquad\text{if}\quad\operatorname{rank}(b)=1,$
$\displaystyle 3.\quad B(G)=\left(\begin{matrix}0&0&b_{1,3}&b_{1,4}\\\
0&0&b_{2,3}&b_{2,4}\\\ b_{1,3}&b_{2,3}&b_{3,3}&b_{3,4}\\\
b_{1,4}&b_{2,4}&b_{3,4}&b_{4,4}\end{matrix}\right)\qquad\text{if}\quad\operatorname{rank}(b)=0.$

Below we analyze the different cases according to $\operatorname{rank}(b)$ we
summarize the results of all cases in Table 3.

###  4.4 The rank 2 case

If

$\displaystyle B({G})=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&b_{3,3}&b_{3,4}\\\
0&0&b_{4,3}&b_{4,4}\end{matrix}\right)\qquad\text{and}\qquad(A,z)\cdot
B(G)=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
0&0&\widetilde{b}_{4,3}&\widetilde{b}_{4,4}\end{matrix}\right),$

for some $(A,z)\in G_{\operatorname{degn}}^{\operatorname{K}}$ then

$\displaystyle\widehat{A}=\left(\begin{matrix}A_{1,1}&A_{1,2}&0&0\\\
A_{2,1}&A_{2,2}&0&0\\\ 0&0&A_{3,3}&0\\\
0&0&0&A_{4,4}\end{matrix}\right),\qquad
z=(A_{1,1}A_{2,2}-A_{1,2}A_{2,1})^{-1}A_{5,5}^{-1},$

with

$\displaystyle\sqrt{z}\left(\begin{matrix}A_{1,1}&A_{1,2}\\\
A_{2,1}&A_{2,2}\end{matrix}\right)\in{\rm O}_{2}(\mathbb{C}),\qquad
A_{5,5}^{-1}=\pm
1,\qquad\left(\begin{matrix}\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
\widetilde{b}_{3,4}&\widetilde{b}_{4,4}\end{matrix}\right)=\left(\begin{matrix}zA_{3,3}^{2}b_{3,3}&zA_{3,3}b_{3,4}\\\
zA_{3,3}b_{3,4}&zb_{4,4}\end{matrix}\right).$

So if $b_{3,3}\neq 0$ we define the canonical form of $B({G})$ to be

$\displaystyle B({G})^{21}(b_{3,4},b_{4,4})=\left(\begin{matrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&b_{3,4}\\\ 0&0&b_{3,4}&b_{4,4}\end{matrix}\right),$

with $b_{3,4}\in\\{0,1\\}$, $b_{4,4}\in{\mathcal{C}}$. If $b_{3,3}=0$ we
define the canonical form of $B({G})$ to be

$\displaystyle B({G})^{22}(b_{3,4},b_{4,4})=\left(\begin{matrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&0&b_{3,4}\\\ 0&0&b_{3,4}&b_{4,4}\end{matrix}\right),$

with $b_{3,4},b_{4,4}\in\\{0,1\\}$.

###  4.5 The rank 1 case

If

$\displaystyle B(G)=\left(\begin{matrix}1&0&0&0\\\ 0&0&b_{2,3}&b_{2,4}\\\
0&b_{2,3}&b_{3,3}&b_{3,4}\\\
0&b_{2,4}&b_{3,4}&b_{4,4}\end{matrix}\right)\qquad\text{and}\qquad(A,z)\cdot
B(G)=\left(\begin{matrix}1&0&0&0\\\
0&0&\widetilde{b}_{2,3}&\widetilde{b}_{2,4}\\\
0&\widetilde{b}_{2,3}&\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
0&\widetilde{b}_{2,4}&\widetilde{b}_{3,4}&\widetilde{b}_{4,4}\end{matrix}\right),$

for some $(A,z)\in G_{\operatorname{degn}}^{\operatorname{K}}$ then

$\displaystyle\widehat{A}=\left(\begin{matrix}\pm{z}^{-1/2}&0&\mp{z}^{1/2}b_{2,3}A_{3,3}A_{2,1}&\mp{z}^{-1/2}b_{2,4}A_{4,4}A_{2,1}\\\
A_{2,1}&A_{2,2}&A_{2,3}&A_{2,4}\\\ 0&0&A_{3,3}&0\\\
0&0&0&A_{4,4}\end{matrix}\right),\qquad z=A_{2,2}^{-2}A_{4,4}^{-1},$

and

$\displaystyle\widetilde{b}_{2,3}=b_{2,3}A_{2,2}^{-1}A_{3,3}A_{4,4}^{-1},$
(4.4) $\displaystyle\widetilde{b}_{2,4}=b_{2,4}A_{2,2}^{-1},$ (4.5)
$\displaystyle\widetilde{b}_{3,3}=b_{2,3}^{2}A_{2,1}^{2}A_{2,2}^{-4}A_{3,3}^{2}A_{4,4}^{-2}+2b_{2,3}A_{2,3}A_{2,2}^{-2}A_{3,3}A_{4,4}^{-1}+b_{3,3}A_{2,2}^{-2}A_{3,3}^{2}A_{4,4}^{-1},$
$\displaystyle\widetilde{b}_{3,4}=b_{2,3}b_{2,4}A_{2,1}^{2}A_{2,2}^{-4}A_{3,3}A_{4,4}^{-1}+b_{2,3}A_{2,4}A_{2,2}^{-2}A_{3,3}A_{4,4}^{-1}+b_{2,4}A_{2,3}A_{2,2}^{-2}+b_{3,4}A_{2,2}^{-2}A_{3,3},$
$\displaystyle\widetilde{b}_{4,4}=b_{2,4}^{2}A_{2,1}^{2}A_{2,2}^{-4}+2b_{2,4}A_{2,4}A_{2,2}^{-2}+b_{4,4}A_{2,2}^{-2}A_{4,4}.$

From (4.4) we see that $\widetilde{b}_{2,3}=0$ if and only if ${b}_{2,3}=0$,
similarly from (4.5) we see that $\widetilde{b}_{2,4}=0$ if and only if
${b}_{2,4}=0$. Hence we continue our analysis according to the vanishing of
$b_{2,3}$ and $b_{2,4}$.

####  4.5.1 $\boldsymbol{\operatorname{rank}(b)=1}$, $\boldsymbol{b_{2,3}=0}$
and $\boldsymbol{b_{2,4}=0}$

If

$\displaystyle B(G)=\left(\begin{matrix}1&0&0&0\\\ 0&0&0&0\\\
0&0&b_{3,3}&b_{3,4}\\\
0&0&b_{3,4}&b_{4,4}\end{matrix}\right)\qquad\text{and}\qquad(A,z)\cdot
B(G)=\left(\begin{matrix}1&0&0&0\\\ 0&0&0&0\\\
0&0&\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
0&0&\widetilde{b}_{3,4}&\widetilde{b}_{4,4}\end{matrix}\right),$

for some $(A,z)\in G_{\operatorname{degn}}^{\operatorname{K}}$ then

$\displaystyle\widehat{A}=\left(\begin{matrix}\pm{z}^{-1/2}&0&0&0\\\
A_{2,1}&A_{2,2}&A_{2,3}&A_{2,4}\\\ 0&0&A_{3,3}&0\\\
0&0&0&A_{4,4}\end{matrix}\right),\qquad z=A_{2,2}^{-2}A_{4,4}^{-1},$

and

$\displaystyle\left(\begin{matrix}\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
\widetilde{b}_{3,4}&\widetilde{b}_{4,4}\end{matrix}\right)=\left(\begin{matrix}A_{2,2}^{-2}A_{4,4}^{-1}A_{3,3}^{2}b_{3,3}&A_{2,2}^{-2}A_{4,4}^{-1}A_{3,3}A_{4,4}b_{3,4}\\\
A_{2,2}^{-2}A_{4,4}^{-1}A_{3,3}A_{4,4}b_{3,4}&A_{2,2}^{-2}A_{4,4}b_{4,4}\end{matrix}\right).$

We define the canonical form to be

$\displaystyle
B(G)^{11}(b_{3,3},b_{3,4},b_{4,4})=\left(\begin{matrix}1&0&0&0\\\ 0&0&0&0\\\
0&0&b_{3,3}&b_{3,4}\\\ 0&0&b_{3,4}&b_{4,4}\end{matrix}\right),$

with $b_{3,3},b_{3,4},b_{4,4}\in\\{0,1\\}$.

####  4.5.2 $\boldsymbol{\operatorname{rank}(b)=1}$, $\boldsymbol{b_{2,3}\neq
0}$ and $\boldsymbol{b_{2,4}\neq 0}$

In this case we can act with $(A,z)\in
G_{\operatorname{degn}}^{\operatorname{K}}$ to obtain

$\displaystyle B(G)^{15}(b_{3,4})=\left(\begin{matrix}1&0&0&0\\\ 0&0&1&1\\\
0&1&0&b_{3,4}\\\ 0&1&b_{3,4}&0\end{matrix}\right),$

for a unique $b_{3,4}\in\\{0,1\\}$. We define this form to be the canonical
form in this case.

####  4.5.3 $\boldsymbol{\operatorname{rank}(b)=1}$, $\boldsymbol{b_{2,3}\neq
0}$ and $\boldsymbol{b_{2,4}=0}$

In this case we can act with $(A,z)\in
G_{\operatorname{degn}}^{\operatorname{K}}$ to obtain

$\displaystyle B(G)^{16}(b_{4,4})=\left(\begin{matrix}1&0&0&0\\\ 0&0&1&0\\\
0&1&0&0\\\ 0&0&0&b_{4,4}\end{matrix}\right),$

with unique $b_{4,4}\in\\{0,1\\}$ which is defined to be the canonical form in
this case.

####  4.5.4 $\boldsymbol{\operatorname{rank}(b)=1}$, $\boldsymbol{b_{2,3}=0}$
and $\boldsymbol{b_{2,4}\neq 0}$

In this case we can act with $(A,z)\in
G_{\operatorname{degn}}^{\operatorname{K}}$ to obtain

$\displaystyle B(G)^{17}(b_{3,3})=\left(\begin{matrix}1&0&0&0\\\ 0&0&0&1\\\
0&0&b_{3,3}&0\\\ 0&1&0&0\end{matrix}\right),$

with unique $b_{3,3}\in\\{0,1\\}$ which is defined to be the canonical form in
this case.

###  4.6 The rank 0 case

If

$\displaystyle B(G)=\left(\begin{matrix}0&0&b_{1,3}&b_{1,4}\\\
0&0&b_{2,3}&b_{2,4}\\\ b_{1,3}&b_{2,3}&b_{3,3}&b_{3,4}\\\
b_{1,4}&b_{2,4}&b_{3,4}&b_{4,4}\end{matrix}\right)\qquad\text{and}\qquad(A,z)\cdot
B(G)=\left(\begin{matrix}0&0&\widetilde{b}_{1,3}&\widetilde{b}_{1,4}\\\
0&0&\widetilde{b}_{2,3}&\widetilde{b}_{2,4}\\\
\widetilde{b}_{1,3}&\widetilde{b}_{2,3}&\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
\widetilde{b}_{1,4}&\widetilde{b}_{2,4}&\widetilde{b}_{3,4}&\widetilde{b}_{4,4}\end{matrix}\right),$

for some $(A,z)\in G_{\operatorname{degn}}^{\operatorname{K}}$ then

$\displaystyle\left(\begin{matrix}\widetilde{b}_{1,3}&\widetilde{b}_{1,4}\\\
\widetilde{b}_{2,4}&\widetilde{b}_{2,4}\end{matrix}\right)=z\left(\begin{matrix}A_{1,1}&A_{2,1}\\\
A_{1,2}&A_{2,2}\end{matrix}\right)\left(\begin{matrix}b_{1,3}&b_{1,4}\\\
b_{2,3}&b_{2,4}\end{matrix}\right)\left(\begin{matrix}A_{3,3}&0\\\
0&A_{4,4}\end{matrix}\right),$

and

$\displaystyle\left(\begin{matrix}\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
\widetilde{b}_{4,3}&\widetilde{b}_{4,4}\end{matrix}\right)=z\left(\begin{matrix}A_{1,3}&A_{2,3}\\\
A_{1,4}&A_{2,4}\end{matrix}\right)\left(\begin{matrix}b_{1,3}&b_{1,4}\\\
b_{2,3}&b_{2,4}\end{matrix}\right)\left(\begin{matrix}A_{3,3}&0\\\
0&A_{4,4}\end{matrix}\right)$
$\displaystyle\hphantom{\left(\begin{matrix}\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
\widetilde{b}_{4,3}&\widetilde{b}_{4,4}\end{matrix}\right)=}{}+z\left(\begin{matrix}A_{3,3}&0\\\
0&A_{4,4}\end{matrix}\right)\left(\begin{matrix}b_{1,3}&b_{2,3}\\\
b_{1,4}&b_{2,4}\end{matrix}\right)\left(\begin{matrix}A_{1,3}&A_{1,4}\\\
A_{2,3}&A_{2,4}\end{matrix}\right)$
$\displaystyle\hphantom{\left(\begin{matrix}\widetilde{b}_{3,3}&\widetilde{b}_{3,4}\\\
\widetilde{b}_{4,3}&\widetilde{b}_{4,4}\end{matrix}\right)=}{}+z\left(\begin{matrix}A_{3,3}&0\\\
0&A_{4,4}\end{matrix}\right)\left(\begin{matrix}b_{3,3}&b_{3,4}\\\
b_{3,4}&b_{4,4}\end{matrix}\right)\left(\begin{matrix}A_{3,3}&0\\\
0&A_{4,4}\end{matrix}\right).$

Hence in the case of $b=0$, the rank of
$c=\left(\begin{smallmatrix}b_{1,3}&b_{1,4}\\\
b_{2,3}&b_{2,4}\end{smallmatrix}\right)$ is invariant under the action of
$G_{\operatorname{degn}}^{\operatorname{K}}$. We continue our analysis
according to the value of this rank. By Corollary 4.4 $c\neq 0$.

####  4.6.1 $\boldsymbol{\operatorname{rank}(b)=0}$,
$\boldsymbol{\operatorname{rank}(c)=1}$

In this case we have the following forms

$\displaystyle\left(\begin{matrix}0&0&0&0\\\ 0&0&1&1\\\ 0&1&b_{3,3}&b_{3,4}\\\
0&1&b_{3,4}&b_{4,4}\end{matrix}\right),\qquad\left(\begin{matrix}0&0&0&0\\\
0&0&1&0\\\ 0&1&b_{3,3}&b_{3,4}\\\
0&0&b_{3,4}&b_{4,4}\end{matrix}\right),\qquad\left(\begin{matrix}0&0&0&0\\\
0&0&0&1\\\ 0&0&b_{3,3}&b_{3,4}\\\ 0&1&b_{3,4}&b_{4,4}\end{matrix}\right).$

The first case can be reduced to the canonical form

$\displaystyle B(G)^{05}=\left(\begin{matrix}0&0&0&0\\\ 0&0&1&1\\\
0&1&0&b_{3,4}\\\ 0&1&b_{3,4}&0\end{matrix}\right),$

for a unique $b_{3,4}\in\\{0,1\\}$. The second case can be reduced to the
canonical form

$\displaystyle B(G)^{06}=\left(\begin{matrix}0&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&b_{4,4}\end{matrix}\right),$

for a unique $b_{4,4}\in\\{0,1\\}$. The case with $b_{4,4}=0$ is not possible
due to Corollary 4.4. The third case can be reduced to the following canonical
form

$\displaystyle B(G)^{07}(b_{3,3})=\left(\begin{matrix}0&0&0&0\\\ 0&0&0&1\\\
0&0&b_{3,3}&0\\\ 0&1&0&0\end{matrix}\right),$

for a unique $b_{3,3}\in\\{0,1\\}$.

####  4.6.2 $\boldsymbol{\operatorname{rank}(b)=0}$,
$\boldsymbol{\operatorname{rank}(c)=2}$

In this case we only have the form

$\displaystyle\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&b_{3,3}&b_{3,4}\\\
0&1&b_{3,4}&b_{4,4}\end{matrix}\right),$ (4.6)

which can be reduced to the canonical form

$\displaystyle B(G)^{08}=\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{matrix}\right).$

degenerate quadratic algebras
---
$\\#$ | $\operatorname{rank}(b)$ | invariant form |  canonical form
1 | 2 | $B({G})^{21}(b_{34},b_{44})$ |  $\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+\mathcal{H}^{2}+b_{44}{\cal X}^{4}+2b_{34}\mathcal{H}{\cal X}^{2}$,2pt $b_{34}\in\\{0,1\\}$, $b_{44}\in\mathbb{C}$2pt
2 | 2 | $B({G})^{22}(b_{34},b_{44})$ |  $\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+b_{44}{\cal X}^{4}+2b_{34}\mathcal{H}{\cal X}^{2}$,2pt $b_{3,4},b_{44}\in\\{0,1\\}$2pt
3 | 1 | $B({G})^{11}(b_{33},b_{34},b_{44})$ |  $\mathcal{L}_{1}^{2}+b_{33}\mathcal{H}^{2}+b_{44}{\cal X}^{4}+2b_{34}\mathcal{H}{\cal X}^{2}$,2pt $b_{33},b_{34},b_{44}\in\\{0,1\\}$, $b_{34}+b_{44}\neq 0$2pt
4 | 1 | $B({G})^{15}(b_{34})$ |  $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}\mathcal{H}+2\mathcal{L}_{2}{\cal X}^{2}+2b_{34}\mathcal{H}{\cal X}^{2}$, $b_{34}\in\\{0,1\\}$ 2pt2pt
5 | 1 | $B({G})^{16}(b_{44})$ |  $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}\mathcal{H}+b_{44}{\cal X}^{4}$, $b_{44}\in\\{0,1\\}$ 2pt2pt
6 | 1 | $B(G)^{17}(b_{33})$ |  $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}\mathcal{X}^{2}+b_{33}{\mathcal{H}}^{2}$, $b_{33}\in\\{0,1\\}$ 2pt2pt
7 | 0 | $B({G})^{05}(b_{34})$ |  $2\mathcal{L}_{2}\mathcal{H}+2\mathcal{L}_{2}{\cal X}^{2}+2b_{34}\mathcal{H}{\cal X}^{2}$, $b_{34}\in\\{0,1\\}$ 2pt2pt
8 | 0 | $B({G})^{06}$ |  $2\mathcal{L}_{2}\mathcal{H}+{\cal X}^{4}$ 2pt2pt
9 | 0 | $B(G)^{07}(b_{33})$ |  $2\mathcal{L}_{2}{\cal X}^{2}+b_{33}\mathcal{H}^{2}$, $b_{33}\in\\{0,1\\}$ 2pt2pt
10 | 0 | $B({G})^{08}$ |  $2\mathcal{L}_{1}\mathcal{H}+2\mathcal{L}_{2}{\cal X}^{2}$ 2pt2pt
Table 3: List of canonical forms of free abstract degenerate quadratic
algebras. The canonical forms $B(G)^{ab}$ are given explicitly in Sections
4.4, 4.5, 4.6 above.

###  4.7 Comparison of geometric and abstract free degenerate quadratic
algebras

We examine the entries in Table 3 to determine which can be realized as a
classical 2D degenerate superintegrable system or a classical Lie algebra with
linearly independent generators. There is a close relationship between the
canonical forms of free abstract degenerate quadratic algebras and Stäckel
equivalence classes of degenerate superintegrable systems. We demonstrate this
by treating one example in detail. The superintegrable system $S3$, with
degenerate potential, can be defined by

$\displaystyle{\mathcal{G}}={\mathcal{L}}_{1}^{2}+{\mathcal{L}}_{2}^{2}-{\mathcal{L}}_{1}{\mathcal{H}}+{\mathcal{L}}_{1}{\mathcal{X}}^{2}+a_{1}{\mathcal{X}}^{2}+(a_{1}+a_{2}){\mathcal{L}}_{1}=0,$

where the $a_{j}$ are the parameters in the potential. To perform a general
Stäckel transform of this system with nonsingular transform matrix
$C=(c_{jk})$: 1) we set $a_{j}=\sum\limits_{k=1}^{2}c_{jk}b_{k}$, $k=1,2$
where the $b_{k}$ are the new parameters, 2) we make the replacements
${\mathcal{H}}\to-b_{2}$, $b_{2}\to-{\mathcal{H}}$ and 3) we then set all
parameters $b_{j}=0$ to determine the free degenerate quadratic algebra. The
result is

$\displaystyle[A]\colon\quad{\mathcal{G}}={\mathcal{L}}_{1}^{2}+{\mathcal{L}}_{2}^{2}+{\mathcal{L}}_{1}{\mathcal{X}}^{2}-c_{12}{\mathcal{H}}{\mathcal{X}}^{2}-(c_{12}+c_{22}){\mathcal{H}}{\mathcal{L}}_{1}=0,$

where $|c_{12}|+|c_{22}|>0$. The canonical forms in Table 3 associated with
the equivalence class $[A]$ are $1$: $b_{44}=1$ and $2$: $b_{44}=1$.

The superintegrable system $E6$, with degenerate potential, can be defined by

$\displaystyle{\mathcal{G}}={\mathcal{L}}_{1}^{2}-{\mathcal{L}}_{2}{\mathcal{H}}+{\mathcal{L}}_{2}{\mathcal{X}}^{2}+a_{1}{\mathcal{X}}^{2}+a_{2}{\mathcal{L}}_{2}=0.$

Going through the same procedure as above, we obtain the equivalence class

$\displaystyle[B]\colon\quad{\mathcal{G}}={\mathcal{L}}_{1}^{2}+{\mathcal{L}}_{2}{\mathcal{X}}^{2}-c_{12}{\mathcal{H}}{\mathcal{X}}^{2}-c_{22}{\mathcal{H}}{\mathcal{L}}_{2}=0,$

where $|c_{12}|+|c_{22}|>0$. The canonical form associated with this
equivalence class is $4$: all cases.

The superintegrable system $E3$, with degenerate potential, can be defined by

$\displaystyle{\mathcal{G}}={\mathcal{L}}_{1}^{2}+{\mathcal{L}}_{2}^{2}-{\mathcal{L}}_{1}{\mathcal{H}}+a_{2}\big{(}{\mathcal{X}}^{2}+a_{2}\big{)}{\mathcal{L}}_{1}.$

The equivalence class is

$\displaystyle[C]\colon\quad{\mathcal{G}}={\mathcal{L}}_{1}^{2}+{\mathcal{L}}_{2}^{2}-c_{12}{\mathcal{H}}{\mathcal{X}}^{2}-c_{22}{\mathcal{H}}{\mathcal{L}}_{1}=0,$

where $|c_{12}|+|c_{22}|>0$. The canonical forms associated with this
equivalence class are $1$: $b_{44}=0$, and $2$: $b_{44}=0$.

The superintegrable system $E5$ can be defined by

$\displaystyle{\mathcal{G}}={\mathcal{L}}_{1}^{2}+{\mathcal{X}}^{4}-{\mathcal{H}}{\mathcal{X}}^{2}+a_{1}{\mathcal{L}}_{2}+a_{2}{\mathcal{X}}^{2}=0.$

The equivalence class is

$\displaystyle[D]\colon\quad{\mathcal{G}}={\mathcal{L}}_{1}^{2}+{\mathcal{X}}^{4}-c_{22}{\mathcal{H}}{\mathcal{X}}^{2}-c_{12}{\mathcal{H}}{\mathcal{L}}_{2}=0,$

where $|c_{12}|+|c_{22}|>0$. The canonical forms associated with this
equivalence class are $3$: $b_{44}=1$, $b_{34}=0$ and $5$: $b_{44}=1$.

The superintegrable system $E14$ can be defined by

$\displaystyle{\mathcal{G}}=-{\mathcal{L}}_{1}^{2}-{\mathcal{L}}_{2}{\mathcal{X}}^{2}+a_{1}{\mathcal{H}}-a_{1}a_{2}=0.$

The equivalence class is

$\displaystyle[E]\colon\quad{\mathcal{G}}^{2}=-{\mathcal{L}}_{1}^{2}-{\mathcal{L}}_{2}{\mathcal{X}}^{2}-c_{12}c_{22}{\mathcal{H}}^{2}=0,$

where $|c_{12}|+|c_{22}|>0$. The canonical forms associated with $[E]$ are
$6$: all cases.

The superintegrable system $E4$ can be defined by

$\displaystyle{\mathcal{G}}={\mathcal{H}}^{2}+{\mathcal{X}}^{4}+2{\mathcal{H}}{\mathcal{X}}^{2}-4{\mathcal{L}}_{2}{\mathcal{X}}^{2}-4ia_{1}{\mathcal{L}}_{1}-2a_{2}{\mathcal{X}}^{2}-2a_{2}{\mathcal{H}}+a_{2}^{2}=0.$

The equivalence class is

$\displaystyle[F]\colon\quad{\mathcal{G}}={\mathcal{X}}^{4}-4{\mathcal{L}}_{2}{\mathcal{X}}^{2}-4ic_{12}{\mathcal{H}}{\mathcal{L}}_{1}+2c_{22}{\mathcal{H}}{\mathcal{X}}^{2}+c_{22}^{2}{\mathcal{H}}^{2}=0,$

where $|c_{12}|+|c_{22}|>0$. The canonical forms associated with $[F]$ are
$9$: $b_{33}=1$ and $10$: all cases.

Heisenberg systems: In addition there are systems that can be obtained from
the degenerate geometric systems above by contractions from
${\mathfrak{so}}(4,\mathbb{C})$ to $\mathfrak{e}(3,\mathbb{C})$. These are not
Bôcher contractions and the contracted systems are not superintegrable,
because the Hamiltonians become singular. However, they do form quadratic
algebras and several have the interpretation of time-dependent Schrödinger
equations in 2D spacetime, so we also consider them geometrical. Some of these
were classified in [16] where they were called Heisenberg systems since they
appeared in quadratic algebras formed from 2nd order elements in the
Heisenberg algebra with generators ${\cal M}_{1}=p_{x}$, ${\cal
M}_{2}=xp_{y}$, ${\cal E}=p_{y}$, where ${\cal E}^{2}={\cal H}$. The systems
are all of type 4. We will devote a future paper to their study. The possible
canonical forms are $3$: $b_{33}=b_{44}=0$, $b_{34}=1$ and $5$: $b_{44}=0$.

rank | canonical form
---|---
2 |  $1$: all cases |  $2$: all cases
1 |  $3$: all cases except |  $4$: all cases
|  $(b_{44},b_{34}=1,b_{33}=0,1)$, $(b_{44}=0,b_{34},b_{33}=1)$ |
1 |  $5$: all cases |  $6$: all cases
0 |  $7$: no |  $8$: missing $2{\mathcal{L}}_{2}{\mathcal{H}}+{\mathcal{X}}^{4}=0$
0 |  $9$: missing ${\mathcal{L}}_{2}{\mathcal{X}}^{2}=0$ |  $10$: all cases
Table 4: Matching of geometric with abstract quadratic algebras.

All these results relating geometric systems to abstract systems are
summarized in Table 4. We see that every abstract quadratic algebra is
isomorphic to a quadratic algebra corresponding to a superintegrable system,
with just 5 exceptions.

######  Theorem 4.6.

Every free quadratic algebra realizable by functions on $4$-dimensional phase
space with grading the order of polynomials in the momenta is isomorphic to a
free quadratic algebra of a superintegrable system.

Proof: We show that the 5 exceptional free quadratic algebras cannot be
realized in phase space. We assume in each case that the algebra is realizable
in terms of functions on phase space and obtain a contradiction.

  1. 1.

Case 3: ${\cal L}_{1}^{2}+\big{(}{\cal H}+{\cal X}^{2}\big{)}^{2}=0$. We can
factor ${\cal G}$ as $({\cal H}+{\cal X}^{2}+i{\cal L}_{1})({\cal H}+{\cal
X}^{2}-i{\cal L}_{1})=0$. This is possible only if one of the factors
vanishes; hence the generators are linearly dependent. Impossible!

  2. 2.

Case 3: ${\cal L}_{1}^{2}+2{\cal H}{\cal X}^{2}+{\cal X}^{4}=0$. Here, ${\cal
L}_{1}^{2}=-{\cal X}^{2}(2{\cal H}+{\cal X}^{2})$ so ${\cal L}_{1}={\cal
X}{\cal Y}$ where ${\cal Y}$ is a 1st order constant of the motion. Thus
${\cal Y}$ must be a multiple of ${\cal X}$ and the generators are linearly
dependent. Impossible!

  3. 3.

Case 3: ${\cal L}_{1}^{2}+{\cal H}^{2}+2{\cal H}{\cal X}^{2}=0$. Here, ${\cal
L}_{1}^{2}=-{\cal H}({\cal H}+2{\cal X}^{2})$. If ${\cal L}_{1}$ doesn’t
factor then it must be a multiple of ${\cal H}$ and of ${\cal H}+2{\cal
X}^{2}$ simultaneously. Impossible! Suppose then that ${\cal L}_{1}={\cal
Y}{\cal Z}$, a product of two 1st order factors. If ${\cal Z}$ is a multiple
of ${\cal Y}$ then ${\cal Y}$ is a constant of the motion, hence proportional
to ${\cal X}$. Impossible! Thus ${\cal Y},{\cal Z}$ must be distinct linear
factors. If ${\cal H}$ is divisible by each factor then ${\cal L}_{1}$, ${\cal
H}$ are linearly dependent. Impossible! So $\cal H$ must be divisible by the
square of a single factor. By renormalizing $\cal Y$ and $\cal Z$ we can
assume ${\cal H}={\cal Y}^{2}$. Thus ${\cal Y}$ is a 1st order constant of the
motion, necessarily proportional to ${\cal X}$. We conclude that ${\cal
H}\sim{\cal X}^{2}$. Impossible!

  4. 4.

Case 8: $2{\cal L}_{2}{\cal H}+{\cal X}^{4}=0$. Since ${\cal X}$ is 1st order,
and $2{\cal L}_{2}{\cal H}=-{\cal X}^{4}$ both ${\cal H}$ and ${\cal L}_{2}$
must be divisible by ${\cal X}^{2}$. Thus each is a perfect square of a 1st
order symmetry, necessarily a scalar multiple of $\cal X$. Hence the 2nd order
generators are linearly dependent. Impossible!

  5. 5.

Case 9: ${\cal L}_{2}{\cal X}=0$. Here at least one of the generators must
vanish. Impossible.!

Thus we have shown that the only free degenerate quadratic algebras that can
be constructed in phase space are those that arise from superintegrability.
The remaining 5 abstract systems must lie in different graded Poisson
algebras.

##  5 Classification of contractions of free abstract quadratic
degenerate 2D superintegrable systems on constant
curvature spaces and Darboux spaces

In this section we define contractions between free abstract quadratic
algebras. Then we list the canonical forms of the Casimirs of free abstract
quadratic algebras that arises as the symmetry algebras of degenerate 2D free
superintegrable systems on constant curvature spaces or Darboux spaces.
Finally using the canonical forms, we classify all possible contractions
relating free abstract quadratic algebras of degenerate 2D superintegrable
systems on constant curvature spaces or Darboux spaces.

###  5.1 Contraction of a free abstract quadratic algebra

######  Definition 5.1.

Let $\mathcal{A}$, $\mathcal{B}$ be free abstract quadratic algebras with
Casimirs $(B({G}_{\mathcal{A}}),K_{\mathcal{A}})$ and
$(B({G}_{\mathcal{B}}),K_{\mathcal{B}})$, respectively. Suppose that there is
a continuous map $\epsilon\longmapsto(A_{\epsilon},z_{\epsilon})$ from some
punctured neighborhood of $0$ in ${\mathbb{C}}^{*}$, the nonzero complex
numbers, into $G_{\operatorname{degn}}$ and such that
$\lim\limits_{\epsilon\longrightarrow
0}(A_{\epsilon},z_{\epsilon})\cdot(b({G}_{\mathcal{A}}),K_{\mathcal{A}})=(B({G}_{\mathcal{B}}),K_{\mathcal{B}})$.
Then we say that $\mathcal{B}$ is a contraction of $\mathcal{A}$.

The meaning of the definition is that in any generating set
$\left\\{\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4}\right\\}$
of $\mathcal{A}$ (satisfying the same assumptions as before) the corresponding
matrix $B({G}_{\mathcal{A}})$ and ${K}_{\mathcal{A}}$ satisfy

$\displaystyle\lim_{\epsilon\longrightarrow
0}z_{\epsilon}\widehat{A_{\epsilon}}^{\rm
t}B({G}_{\mathcal{A}})\widehat{A_{\epsilon}}=B({G}_{\mathcal{B}}),$ (5.1)
$\displaystyle\lim_{\epsilon\longrightarrow
0}z_{\epsilon}^{-1}K_{\mathcal{A}}\left((A_{\epsilon})_{1,1}(A_{\epsilon})_{2,2}-(A_{\epsilon})_{1,2}(A_{\epsilon})_{2,1}\right)^{-1}(A_{\epsilon})_{5,5}^{-1}=K_{\mathcal{B}},$
(5.2)

where $(B({G}_{\mathcal{B}}),K_{\mathcal{B}})$ is a realization of the Casimir
of $\mathcal{B}$ in some generating set. In the classification below we are
using a more refined class of contractions. We shall call these contractions
algebraic. By definition an algebraic contraction is a contraction that can be
realized via a map $\epsilon\longmapsto(A_{\epsilon},z_{\epsilon})$ from some
punctured neighborhood of $0$ in ${\mathbb{C}}^{*}$ into
$G_{\operatorname{degn}}$ such that $z_{\epsilon}$ as well as the entries of
$A_{\epsilon}$ are rational functions in $\epsilon$.

######  Proposition 5.2.

$\mathcal{B}$ is a contraction of $\mathcal{A}$ if and only if there is a
continuous map $\epsilon\longmapsto(A_{\epsilon},z_{\epsilon})$ from some
punctured neighborhood of $0$ in ${\mathbb{C}}^{*}$ into
$G_{\operatorname{degn}}$ such that

$\displaystyle\lim_{\epsilon\longrightarrow
0}z_{\epsilon}\widehat{A_{\epsilon}}^{\rm t}B({G}_{\mathcal{A}})_{\rm
can}\widehat{A_{\epsilon}}=B({G}_{\mathcal{B}})_{\rm can},$ (5.3)
$\displaystyle\lim_{\epsilon\longrightarrow
0}z_{\epsilon}^{-1}\left((A_{\epsilon})_{1,1}(A_{\epsilon})_{2,2}-(A_{\epsilon})_{1,2}(A_{\epsilon})_{2,1},\right)^{-1}(A_{\epsilon})_{5,5}^{-1}=1,$
(5.4)

where $(B({G}_{\mathcal{A}})_{\rm can},1)$ and $(B({G}_{\mathcal{B}})_{\rm
can},1)$ are the canonical forms of the Casimirs of $\mathcal{A}$ and
$\mathcal{B}$ respectively.

###### Proof.

Obviously if equations (5.3)–(5.4) hold then $\mathcal{B}$ is a contraction of
$\mathcal{A}$. For the other direction assume that equations (5.1)–(5.2) hold.
We can further assume that $(B({G}_{\mathcal{A}}),K_{\mathcal{A}})$ is in its
canonical form $(B({G}_{\mathcal{A}})_{\rm can},1)$. Let $(A,z)\in
G_{\operatorname{degn}}$ such that
$(B({G}_{\mathcal{B}}),K_{\mathcal{B}})=(A,z)\cdot(B({G}_{\mathcal{B}})_{\rm
can},1)$, then the continuity of the action of $G_{\operatorname{degn}}$
implies that $\lim\limits_{\epsilon\longrightarrow
0}\left((A,z)^{-1}(A_{\epsilon},z_{\epsilon})\right)\cdot(B({G}_{\mathcal{A}})_{\rm
can},1)=(B({G}_{\mathcal{B}})_{\rm can},1)$. ∎

######  Proposition 5.3.

Let $\mathcal{A}$, $\mathcal{B}$ be free abstract quadratic algebras and
$\epsilon\longmapsto(A_{\epsilon},1)$ a continuous map from ${\mathbb{C}}^{*}$
into $G_{\operatorname{degn}}$ such that

$\displaystyle\lim_{\epsilon\longrightarrow 0}\widehat{A_{\epsilon}}^{\rm
t}B({G}_{\mathcal{A}})_{\rm
can}\widehat{A_{\epsilon}}=B({G}_{\mathcal{B}})_{\rm can}.$

Then there is another continuous map
$\epsilon\longmapsto({C}_{\epsilon},z_{\epsilon})$ from ${\mathbb{C}}^{*}$
into $G_{\operatorname{degn}}$ such that $\lim\limits_{\epsilon\longrightarrow
0}(C_{\epsilon},z_{\epsilon})\cdot(B({G}_{\mathcal{A}})_{\rm
can},1)=(B({G}_{\mathcal{B}})_{\rm can},1)$.

###### Proof.

Define

$\displaystyle
z_{\epsilon}:=\left((A_{\epsilon})_{1,1}(A_{\epsilon})_{2,2}-(A_{\epsilon})_{1,2}(A_{\epsilon})_{2,1}\right)^{4}(A_{\epsilon})_{5,5}^{4},$
$\displaystyle
C_{\epsilon}=\left((A_{\epsilon})_{1,1}(A_{\epsilon})_{2,2}-(A_{\epsilon})_{1,2}(A_{\epsilon})_{2,1}\right)^{-1}(A_{\epsilon})_{5,5}^{-1}A_{\epsilon}.$
∎

Note that Propositions 5.2 and 5.3 also hold for algebraic contractions. The
conclusion from the last two proposition is that for the purpose of
classifying contractions of free abstract quadratic algebras it is enough to
take the Casimirs $B(G)_{\mathcal{A}}$ and $B(G)_{\mathcal{B}}$ in their
canonical forms and to consider only the action of the group of invertible
matrices of the form

$\displaystyle\widehat{A}=\left(\begin{matrix}A_{1,1}&A_{1,2}&A_{1,3}&A_{1,4}\\\
A_{2,1}&A_{2,2}&A_{2,3}&A_{2,4}\\\ 0&0&A_{3,3}&0\\\
0&0&0&A_{4,4}\end{matrix}\right),$ (5.5)

on symmetric matrices $B$ by $B\longmapsto\widehat{A}^{\rm t}B\widehat{A}$. We
shall denote the space of 4 by 4 complex symmetric matrices by
$\operatorname{Sym}(4,{\mathbb{C}})$ and the group of matrices of the form
(5.5) by $\widehat{G}_{\operatorname{degn}}$. The group
$\widehat{G}_{\operatorname{degn}}$ is a complex algebraic group and the space
${\rm Sym}(4,{\mathbb{C}})$ is a complex algebraic variety on which
$\widehat{G}_{\operatorname{degn}}$ acts algebraically. As was explained in
[5, Section 7.1.2] this implies that if $\mathcal{B}$ is a contraction of a
quadratic algebra $\mathcal{A}$ then $\mathcal{A}$ is not a contraction of
$\mathcal{B}$ (unless $\mathcal{A}$ and $\mathcal{B}$ are isomorphic). In
addition if $\mathcal{B}$ is a contraction of $\mathcal{A}$ then the rank of
any matrix that represents $B(G)_{\mathcal{B}}$ and the rank of its upper left
2 by 2 block can not exceed the corresponding ranks for any matrix that
represents $B(G)_{\mathcal{A}}$. Hence there is certain hierarchy for
contractions that is governed by the rank. By a rank of a free abstract
quadratic algebra we mean $(\operatorname{rank}(B),\operatorname{rank}(b))$ of
the corresponding matrices $B$ and $b$ in any bases. We shall make this more
precise below.

###  5.2 Classification of abstract algebraic contractions
of superintegrable systems

Organized according to their rank, the canonical forms of the free abstract
quadratic algebras of free triplets of 2D constant curvature spaces and
Darboux spaces are given in Table 5 below.

canonical forms for the Casimirs of free degenerate 2D 2nd order
---
superintegrable systems on constant curvature and Darboux spaces
$\\#$ | system | $\operatorname{rank}(B)$ | $\operatorname{rank}(b)$ |  canonical form
1 | $S6$ | 4 | 2 |  $B^{22}(1,1)$, $\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+2\mathcal{H}{\cal X}^{2}+{\cal X}^{4}$2pt2pt
2 | $E18$ | 4 | 2 |  $B^{22}(0,1)$, $\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+2\mathcal{H}{\cal X}^{2}$2pt2pt
3 | $D3E$ | 4 | 2 |  $B^{21}(1,0)$, $\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+{\mathcal{H}}^{2}+2\mathcal{H}{\cal X}^{2}$ 2pt2pt
4 | $D4(b)D$ | 4 | 2 |  $B^{21}(1,-2)$, $\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+\mathcal{H}^{2}+2\mathcal{H}{\cal X}^{2}-2{\cal X}^{4}$
5 | $S3$ | 4 | 2 |  $B^{21}(\sqrt{2}e^{i\frac{3\pi}{4}},-2i)$,2pt2pt $\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+\mathcal{H}^{2}+2\sqrt{2}e^{i\frac{3\pi}{4}}\mathcal{H}{\cal X}^{2}-2i{\cal X}^{4}$
6 | $E3$ | 3 | 2 |  $B^{21}(0,0)$, $\mathcal{L}_{1}^{2}+\mathcal{L}_{2}^{2}+\mathcal{H}^{2}$ 2pt2pt
7 | $E12$ | 4 | 1 |  $B^{17}(1)$, $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}{\cal X}^{2}+{\mathcal{H}}^{2}$ 2pt2pt
8 | $D1D$ | 4 | 1 |  $B^{16}(1)$, $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}\mathcal{H}+\mathcal{H}^{2}$ 2pt2pt
9 | $D2D$ | 4 | 1 |  $B^{15}(1)$, $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}\mathcal{H}+2\mathcal{L}_{2}\mathcal{X}^{2}+2\mathcal{H}\mathcal{X}^{2}$ 2pt2pt
10 | $E6$ | 3 | 1 |  $B^{15}(0)$, $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}{\mathcal{H}}+2\mathcal{L}_{2}{\cal X}^{2}$2pt2pt
11 | $E5$ | 3 | 1 |  $B^{11}(0,0,1)$, $\mathcal{L}_{1}^{2}+2\mathcal{H}{\cal X}^{2}+{\cal X}^{4}$ 2pt2pt
12 | $E14$ | 3 | 1 |  $B^{17}(0)$, $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}{\cal X}^{2}$ 2pt2pt
13 | $S5$ | 3 | 1 |  $B^{17}(0)$, $\mathcal{L}_{1}^{2}+2\mathcal{L}_{2}{\cal X}^{2}$ 2pt2pt
14 | $E13$ | 4 | 0 |  $B^{08}$, $2\mathcal{L}_{1}\mathcal{H}+2\mathcal{L}_{2}\mathcal{X}^{2}$ 2pt2pt
15 | $E4$ | 3 | 0 |  $B^{07}(1)$, $\mathcal{H}^{2}+2\mathcal{L}_{2}\mathcal{X}^{2}$ 2pt2pt
Table 5: List of canonical forms of free abstract degenerate quadratic
algebras.

We shall classify all possible algebraic contractions between any two free
abstract quadratic algebras that appear in Table 5. More precisely for any two
such algebras we determine if such a contraction is possible or not. If it
does we give one realization of it, unless it is a contraction from a free
abstract quadratic algebra to itself. We shall start with some general
observations. We note that the quadratic algebras of ${E}14$ and $S5$ coincide
so we only keep $E14$ in our notation. We divide the free abstract quadratic
algebras of the second order free degenerate superintegrable systems on 2D
constant curvature spaces and 2D Darboux spaces according to their ranks:
$R_{4,2}:=\\{{S}6,{E}18,{D}3E,{D}4(b)D,{S}3\\}$, $R_{3,2}:=\\{{E}3\\}$,
$R_{4,1}:=\\{{E}12,{D}1D,{D}2D\\}$, $R_{3,1}:=\\{{E}6,{E}5,{E}14\\}$,
$R_{4,0}:=\\{{E}13\\}$, $R_{3,0}:=\\{{E}4\\}$. By the discussion above,
besides contractions between two algebras in the same class $R_{i,j}$ we can
potentially find a contraction only according to the following diagram:

$\textstyle{R_{4,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{3,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{4,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{3,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{4,0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{3,0}.}$

The classification is summarized in Table 6. Detailed analysis is presented
below. Representatives for all possible contractions are given in Section
5.3.7.

| ${S}6$ | ${E}18$ | ${D}3E$ | ${D}4(b)D$ | ${S}3$ | ${E}3$ | ${E}12$ | ${D}1D$ | ${D}2D$ | ${E}6$ | ${E}5$ | ${E}14$ | ${E}13$ | ${E}4$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
${S}6$ | + | + | – | – | – | – | + | – | – | – | + | + | + | +
${E}18$ | – | + | – | – | – | – | – | – | – | – | + | – | + | +
${D}3E$ | – | + | + | – | – | + | – | + | – | – | + | – | + | +
${D}4(b)D$ | – | – | – | + | – | + | + | + | – | – | + | + | + | +
${S}3$ | – | – | – | – | + | + | – | + | – | + | + | + | + | +
${E}3$ | – | – | – | – | – | + | – | – | – | – | + | – | – | +
${E}12$ | – | – | – | – | – | – | + | – | – | – | + | + | + | +
${D}1D$ | – | – | – | – | – | – | – | + | – | – | + | – | + | +
${D}2D$ | – | – | – | – | – | – | + | + | + | + | + | + | + | +
${E}6$ | – | – | – | – | – | – | – | – | – | + | + | + | – | +
${E}5$ | – | – | – | – | – | – | – | – | – | – | + | – | – | +
${E}14$ | – | – | – | – | – | – | – | – | – | – | + | + | – | +
${E}13$ | – | – | – | – | – | – | – | – | – | – | – | – | + | +
${E}4$ | – | – | – | – | – | – | – | – | – | – | – | – | – | +
Table 6: List of algebraic contractions between free abstract quadratic
algebras of 2D free degenerate superintegrable systems on constant curvature
spaces and Darboux spaces. A plus in the rubric placed in the $i$-th row and
$j$-th column indicates that there is a contraction from the system listed in
the $i$-th row of the first column to the system listed in the $j$-th column
of the first row. A minus indicates that there is no such contraction.

####  5.2.1 Contractions between rank two free abstract quadratic algebras

Note that for any of the $B_{\rm can}$ matrices for the systems $R_{4,2}\cup
R_{3,2}=\\{{S}6,{E}18,{D}3E,{D}4(b)D,{S}3\\}$ $\cup\\{{E}3\\}$ we have

$\displaystyle\widehat{A}^{\rm t}B\widehat{A}=\left(\begin{matrix}r^{\rm
t}&0\\\ s^{\rm t}&t\end{matrix}\right)\left(\begin{matrix}1&0\\\
0&d\end{matrix}\right)\left(\begin{matrix}r&s\\\
0&t\end{matrix}\right)=\left(\begin{matrix}r^{\rm t}r&r^{\rm t}s\\\ s^{\rm
t}r&s^{\rm t}s+tdt\end{matrix}\right).$ (5.6)

Consider $A=A(\epsilon)$ in equation (5.6) such that

$\displaystyle\lim_{\epsilon\longrightarrow 0}\widehat{A(\epsilon)}^{\rm
t}B\widehat{A(\epsilon)}=B^{0},$ (5.7)

where $B^{0}\in R_{4,2}\cup R_{3,2}$. Below we prove that without loss of
generality we can assume that $r^{\rm t}_{\epsilon}r_{\epsilon}$ is diagonal
and $s_{\epsilon}$ is the zero two by two matrix.

######  Proposition 5.4.

Suppose that $\\{B_{\epsilon}\\}_{\epsilon\in\mathbb{R}^{+}}$ is a family of
$4\times 4$ symmetric matrices with entries in ${\mathbb{C}}[\epsilon]$ and
such that

$\displaystyle\lim_{\epsilon\longrightarrow
0^{+}}\left(\begin{matrix}({B}_{\epsilon})_{11}&({B}_{\epsilon})_{12}&({B}_{\epsilon})_{13}&({B}_{\epsilon})_{14}\\\
({B}_{\epsilon})_{12}&({B}_{\epsilon})_{22}&({B}_{\epsilon})_{23}&({B}_{\epsilon})_{24}\\\
({B}_{\epsilon})_{13}&({B}_{\epsilon})_{23}&({B}_{\epsilon})_{33}&({B}_{\epsilon})_{34}\\\
({B}_{\epsilon})_{14}&({B}_{\epsilon})_{24}&({B}_{\epsilon})_{34}&({B}_{\epsilon})_{44}\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&l_{33}&l_{34}\\\ 0&0&l_{34}&l_{44}\end{matrix}\right).$

Then there exists a continuous function $\epsilon\mapsto A_{\epsilon}$ from
${\mathbb{C}}^{*}$ to $\widehat{G}_{\operatorname{degn}}$ with each entry
polynomial in $\epsilon$ such that $\widehat{A}_{\epsilon}^{\rm
t}B_{\epsilon}\widehat{A}_{\epsilon}$ is of the form

$\displaystyle\left(\begin{matrix}(\widetilde{B}_{\epsilon})_{11}&0&0&0\\\
0&(\widetilde{B}_{\epsilon})_{22}&0&0\\\
0&0&(\widetilde{B}_{\epsilon})_{33}&(\widetilde{B}_{\epsilon})_{34}\\\
0&0&(\widetilde{B}_{\epsilon})_{34}&(\widetilde{B}_{\epsilon})_{44}\end{matrix}\right),$

and

$\displaystyle\lim_{\epsilon\longrightarrow 0}\widehat{A}_{\epsilon}^{\rm
t}B_{\epsilon}\widehat{A}_{\epsilon}=\lim_{\epsilon\longrightarrow
0}B_{\epsilon}.$

###### Proof.

Note that for $\widehat{A}_{\epsilon}^{\rm
t}=\left(\begin{smallmatrix}1&0&0&0\\\
-({B}_{\epsilon})_{12}&({B}_{\epsilon})_{11}&0&0\\\
-({B}_{\epsilon})_{13}&0&({B}_{\epsilon})_{11}&0\\\
-({B}_{\epsilon})_{14}&0&0&({B}_{\epsilon})_{11}\end{smallmatrix}\right)$
$\lim\limits_{\epsilon\longrightarrow 0}\widehat{A}_{\epsilon}^{\rm t}$ is the
identity matrix and $\widehat{A}_{\epsilon}^{\rm
t}B_{\epsilon}\widehat{A}_{\epsilon}$ is a symmetric matrix with all entries
in the first row and first column besides the $(1,1)$ entry equal to zero.
Similar matrix take care of the second row and second column. ∎

Now suppose that equation (5.7) holds. Using the last proposition this means
that we can assume that
$\widehat{A(\epsilon)}=\left(\begin{smallmatrix}r_{\epsilon}&0\\\
0&t_{\epsilon}\end{smallmatrix}\right)$ with $r_{\epsilon}\in{\rm
GL}(2,{\mathbb{C}})$ and, as always, $t_{\epsilon}$ a diagonal invertible
matrix. From this we easily see that the only possible contractions between
two algebras in $R_{4,2}\cup R_{3,2}$ are $S6\longrightarrow E18$,
$D3E\longrightarrow E18$, $D3E\longrightarrow E3$, $D4(b)D\longrightarrow E3$,
$S3\longrightarrow E3$.

###  5.3 Contractions of rank two free abstract quadratic algebras
to rank one algebras

######  Proposition 5.5.

Suppose that $\\{B_{\epsilon}\\}_{\epsilon\in\mathbb{R}^{+}}$ is a family of
$4\times 4$ symmetric matrices with entries in ${\mathbb{C}}[\epsilon]$ such
that

$\displaystyle\lim_{\epsilon\longrightarrow
0^{+}}B_{\epsilon}=L=\left(\begin{matrix}1&0&l_{13}&l_{14}\\\
0&0&l_{23}&l_{24}\\\ l_{13}&l_{23}&l_{33}&l_{34}\\\
l_{14}&l_{24}&l_{34}&l_{44}\end{matrix}\right).$

Then there exists a continuous function $\epsilon\mapsto A_{\epsilon}$ from
${\mathbb{C}}^{*}$ to $\widehat{G}_{\operatorname{degn}}$ with each entry
polynomial in $\epsilon$ such that $\widehat{A}_{\epsilon}^{\rm
t}B_{\epsilon}\widehat{A}_{\epsilon}$ is of the form

$\displaystyle\left(\begin{matrix}(\widetilde{B}_{\epsilon})_{11}&0&0&0\\\
0&(\widetilde{B}_{\epsilon})_{22}&(\widetilde{B}_{\epsilon})_{23}&(\widetilde{B}_{\epsilon})_{24}\\\
0&(\widetilde{B}_{\epsilon})_{23}&(\widetilde{B}_{\epsilon})_{33}&(\widetilde{B}_{\epsilon})_{34}\\\
0&(\widetilde{B}_{\epsilon})_{24}&(\widetilde{B}_{\epsilon})_{34}&(\widetilde{B}_{\epsilon})_{44}\end{matrix}\right),$

and

$\displaystyle\lim_{\epsilon\longrightarrow 0}\widehat{A}_{\epsilon}^{\rm
t}B_{\epsilon}\widehat{A}_{\epsilon}=\lim_{\epsilon\longrightarrow
0}B_{\epsilon}.$

If in addition $l_{13}=0$ then $(\widetilde{B}_{\epsilon})_{13}=0$ and
similarly if $l_{14}=0$ then $(\widetilde{B}_{\epsilon})_{14}=0$.

The proof is similar to the proof of Proposition 5.4.

######  Corollary 5.6.

In any contraction from a rank two systems $R_{4,2}\cup
R_{3,2}=\\{{S}6,{E}18,{D}3E,$ ${D}4(b)D,{S}3,{E}3\\}$ to of one of the rank
one systems $R_{4,1}\cup R_{3,1}=\\{{E}12,{D}1D,{D}2D,{E}6,{E}5,{E}14\\}$ we
can assume that

$\displaystyle\widehat{A(\epsilon)}^{\rm
t}B\widehat{A(\epsilon)}=\left(\begin{matrix}({B}_{\epsilon})_{11}&0&0&0\\\
0&({B}_{\epsilon})_{22}&({B}_{\epsilon})_{23}&({B}_{\epsilon})_{24}\\\
0&({B}_{\epsilon})_{23}&({B}_{\epsilon})_{33}&({B}_{\epsilon})_{34}\\\
0&({B}_{\epsilon})_{24}&({B}_{\epsilon})_{34}&({B}_{\epsilon})_{44}\end{matrix}\right).$
(5.8)

####  5.3.1 Contractions of $\boldsymbol{D3E}$ to rank one algebras

For $A(\epsilon)\in\widehat{G}_{\operatorname{degn}}$ the matrix
$\widehat{A(\epsilon)}^{\rm t}B({G})^{21}(1,0)\widehat{A(\epsilon)}$ is given
by

$\displaystyle\left(\begin{smallmatrix}A_{11}^{2}+A_{21}^{2}&A_{11}A_{12}+A_{21}A_{22}&A_{11}A_{13}+A_{21}A_{23}&A_{11}A_{14}+A_{21}A_{24}\\\
A_{12}A_{11}+A_{22}A_{21}&A_{12}^{2}+A_{22}^{2}&A_{12}A_{13}+A_{22}A_{23}&A_{12}A_{14}+A_{22}A_{24}\\\
A_{11}A_{13}+A_{21}A_{23}&A_{12}A_{13}+A_{22}A_{23}&A_{13}^{2}+A_{23}^{2}+A_{33}^{2}&A_{13}A_{14}+A_{23}A_{24}+A_{33}A_{44}\\\
A_{11}A_{14}+A_{21}A_{24}&A_{12}A_{14}+A_{22}A_{24}&A_{13}A_{14}+A_{23}A_{24}+A_{33}A_{44}&A_{14}^{2}+A_{24}^{2}\end{smallmatrix}\right).$

Assuming that this matrix is in the form of (5.8) then this implies that exist
$\beta$, $\gamma$, $\delta$ complex valued functions of $\epsilon$ define on
${\mathbb{C}}^{*}$ such that

$\displaystyle(A_{12},A_{22})=\beta(A_{21},-A_{11}),\\!\qquad(A_{13},A_{23})=\gamma(A_{21},-A_{11}),\\!\qquad(A_{14},A_{24})=\delta(A_{21},-A_{11}).$

Hence $\widehat{A(\epsilon)}^{\rm t}B({G})^{21}(1,0)\widehat{A(\epsilon)}$
takes the form

$\displaystyle\left(\begin{matrix}A_{11}^{2}+A_{21}^{2}&0&0&0\\\
0&\beta^{2}(A_{11}^{2}+A_{21}^{2})&\beta\gamma(A_{11}^{2}+A_{21}^{2})&\beta\delta(A_{11}^{2}+A_{21}^{2})\\\
0&\beta\gamma(A_{11}^{2}+A_{21}^{2})&\gamma^{2}(A_{11}^{2}+A_{21}^{2})+A_{33}^{2}&\gamma\delta(A_{11}^{2}+A_{21}^{2})+A_{33}A_{44}\\\
0&\beta\delta(A_{11}^{2}+A_{21}^{2})&\gamma\delta(A_{11}^{2}+A_{21}^{2})+A_{33}A_{44}&\delta^{2}(A_{11}^{2}+A_{21}^{2})\end{matrix}\right).$

We are going to consider the limit of $\epsilon$ goes to zero of such a matrix
and check if it can be equal to one of the matrices of the canonical forms of
the systems in $R_{4,1}\cup R_{3,1}$. This means that
$\lim\limits_{\epsilon\longrightarrow 0^{+}}(A_{11}^{2}+A_{21}^{2})=1$ and we
can replace $\widehat{A(\epsilon)}^{\rm
t}B({G})^{21}(1,0)\widehat{A(\epsilon)}$ by

$\displaystyle\left(\begin{matrix}1&0&0&0\\\
0&\beta^{2}&\beta\gamma&\beta\delta\\\
0&\beta\gamma&\gamma^{2}+A_{33}^{2}&\gamma\delta+A_{33}A_{44}\\\
0&\beta\delta&\gamma\delta+A_{33}A_{44}&\delta^{2}\end{matrix}\right),$

with $\lim\limits_{\epsilon\longrightarrow 0^{+}}\beta=0$. From this we can
show that $D3E$ can not be contracted to $E6$, $E12$, $E14$, $D2D$.

####  5.3.2 Contractions of $\boldsymbol{E18}$ to rank one algebras

Following the same steps as in the case of $D3E$ we can replace
$\widehat{A(\epsilon)}^{\rm t}B({G})^{22}(0,1)\widehat{A(\epsilon)}$ by

$\displaystyle\left(\begin{matrix}1&0&0&0\\\
0&\beta^{2}&\beta\gamma&\beta\delta\\\
0&\beta\gamma&\gamma^{2}&\gamma\delta+A_{33}A_{44}\\\
0&\beta\delta&\gamma\delta+A_{33}A_{44}&\delta^{2}\end{matrix}\right),$

with $\lim\limits_{\epsilon\longrightarrow 0^{+}}\beta=0$. From this we can
show that $E18$ can not be contracted to $E6$, $E12$, $D1D$, $D2D$.

####  5.3.3 Contractions of $\boldsymbol{S6}$ to rank one algebras

Following the same steps as in the case of $D3E$ we can replace
$\widehat{A(\epsilon)}^{\rm t}B({G})^{22}(1,1)\widehat{A(\epsilon)}$ by

$\displaystyle\left(\begin{matrix}1&0&0&0\\\
0&\beta^{2}&\beta\gamma&\beta\delta\\\
0&\beta\gamma&\gamma^{2}&\gamma\delta+A_{33}A_{44}\\\
0&\beta\delta&\gamma\delta+A_{33}A_{44}&\delta^{2}+A_{44}^{2}\end{matrix}\right),$

with $\lim\limits_{\epsilon\longrightarrow 0^{+}}\beta=0$. From this we can
show that $S6$ can not be contracted to $E6$, $D1D$, $D2D$.

####  5.3.4 Contractions of $\boldsymbol{D4(b)D}$ to rank one algebras

Following the same steps as in the case of $D3E$ we can replace
$\widehat{A(\epsilon)}^{\rm t}B({G})^{21}(1,-2)\widehat{A(\epsilon)}$ by

$\displaystyle\left(\begin{matrix}1&0&0&0\\\
0&\beta^{2}&\beta\gamma&\beta\delta\\\
0&\beta\gamma&\gamma^{2}+A_{33}^{2}&\gamma\delta+A_{33}A_{44}\\\
0&\beta\delta&\gamma\delta+A_{33}A_{44}&\delta^{2}-2A_{44}^{2}\end{matrix}\right),$

with $\lim\limits_{\epsilon\longrightarrow 0^{+}}\beta=0$. From this we can
show that $D4(b)D$ can not be contracted to $D2D$, $E6$.

####  5.3.5 Contractions of $\boldsymbol{S3}$ to rank one algebras

Following the same steps as in the case of $D3E$ we can replace
$\widehat{A(\epsilon)}^{\rm
t}B({G})^{21}(\sqrt{2}e^{i\frac{3\pi}{4}},-2i)\widehat{A(\epsilon)}$ by

$\displaystyle\left(\begin{matrix}1&0&0&0\\\
0&\beta^{2}&\beta\gamma&\beta\delta\\\
0&\beta\gamma&\gamma^{2}+A_{33}^{2}&\gamma\delta+\sqrt{2}e^{i\frac{3\pi}{4}}A_{33}A_{44}\\\
0&\beta\delta&\gamma\delta+\sqrt{2}e^{i\frac{3\pi}{4}}A_{33}A_{44}&\delta^{2}-2iA_{44}^{2}\end{matrix}\right),$

with $\lim\limits_{\epsilon\longrightarrow 0^{+}}\beta=0$. From this we can
show that $S3$ can not be contracted to $D2D$, $E12$.

####  5.3.6 Contractions of $\boldsymbol{E3}$ to rank one algebras

Following the same steps as in the case of $D3E$ we can replace
$\widehat{A(\epsilon)}^{\rm t}B({G})^{21}(0,0)\widehat{A(\epsilon)}$ by

$\displaystyle\left(\begin{matrix}1&0&0&0\\\
0&\beta^{2}&\beta\gamma&\beta\delta\\\
0&\beta\gamma&\gamma^{2}&\gamma\delta\\\
0&\beta\delta&\gamma\delta&\delta^{2}\end{matrix}\right),$

with $\lim\limits_{\epsilon\longrightarrow 0^{+}}\beta=0$. From this we can
show that $E3$ can not be contracted to $E6$ and $E14$.

####  5.3.7 Explicit contractions

E13:

$\displaystyle\begin{array}[]{@{}c@{}}{E}13\longrightarrow{E}4\\\
\left(\begin{matrix}\epsilon&0&\frac{1}{2}&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\right).\end{array}$

E14:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}}{E}14\longrightarrow{E}4&&{E}14\longrightarrow{E}5\\\
\left(\begin{matrix}\epsilon&0&1&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&\epsilon&1&\frac{1}{2}\\\ 0&0&1&0\\\ 0&0&0&1\end{matrix}\right).\end{array}$

E5:

$\displaystyle\begin{array}[]{@{}c@{}}{E}5\longrightarrow{E}4\\\
\left(\begin{matrix}\epsilon^{2}&\epsilon&1&\epsilon^{-1}\\\ 0&1&0&0\\\
0&0&i&0\\\ 0&0&0&i\epsilon^{-1}\end{matrix}\right).\end{array}$

D2D:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{D}2D\longrightarrow{E}6&&{D}2D\longrightarrow{E}12&&{D}2D\longrightarrow{D}1D\\\
\left(\begin{matrix}1&0&1&0\\\ 0&\epsilon^{-1}&0&0\\\ 0&0&\epsilon&0\\\
0&0&0&\epsilon\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&\epsilon&-\frac{i}{\sqrt{2}}&0\\\ 0&0&\frac{i}{\sqrt{2}}&0\\\
0&0&0&\epsilon^{-1}\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&\epsilon&\frac{\epsilon}{2}&0\\\ 0&0&\epsilon^{-1}&0\\\
0&0&0&\epsilon^{2}\end{matrix}\right),\end{array}$
$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}{D}2D\longrightarrow{E}5&&{D}2D\longrightarrow{E}14&&{D}2D\longrightarrow{E}4&&{D}2D\longrightarrow{E}13\\\
\left(\begin{matrix}1&0&0&0\\\ 0&\epsilon&0&\frac{1}{2\epsilon}\\\
0&0&2\epsilon&0\\\
0&0&0&\epsilon\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\
0&0&0&\epsilon\end{matrix}\right),&&\left(\begin{matrix}\epsilon&0&1&1\\\
0&\epsilon&0&0\\\ 0&0&\epsilon&0\\\
0&0&0&\epsilon\end{matrix}\right),&&\left(\begin{matrix}0&\epsilon&0&\epsilon^{-1}\\\
1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&i\epsilon^{-1}\end{matrix}\right).\end{array}$

E6:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{E}6\longrightarrow{E}14&&{E}6\longrightarrow{E}5&&{E}6\longrightarrow{E}4\\\
\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&\epsilon&0\\\
0&0&0&1\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&\epsilon&\epsilon&\epsilon^{-1}\\\ 0&0&\epsilon&0\\\
0&0&0&\frac{1}{2}\epsilon\end{matrix}\right),&&\left(\begin{matrix}\epsilon&0&1&0\\\
0&1&0&0\\\ 0&0&\epsilon&0\\\ 0&0&0&1\end{matrix}\right).\end{array}$

E12:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}{E}12\longrightarrow{E}14&&{E}12\longrightarrow{E}5&&{E}12\longrightarrow{E}4&&{E}12\longrightarrow{E}13\\\
\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&\epsilon&0\\\
0&0&0&1\end{matrix}\right),&&\left(\begin{matrix}1&0&0&1\\\ -1&\epsilon&1&0\\\
0&0&\epsilon&0\\\
0&0&0&1\end{matrix}\right),&&\left(\begin{matrix}\epsilon&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\
0&0&0&1\end{matrix}\right),&&\left(\begin{matrix}\epsilon&0&\epsilon^{-1}&0\\\
0&1&0&0\\\ 0&0&i\epsilon^{-1}&0\\\ 0&0&0&1\end{matrix}\right).\end{array}$

The non-contraction of ${E}12$ to ${D}1D$, $E6$, ${D}2D$: Demanding that
$\lim\limits_{\epsilon\longrightarrow 0}\widehat{A(\epsilon)}^{\rm
t}B^{17}(1)\widehat{A(\epsilon)}$ converge to the Canonical form of one of the
systems ${D}1D$, $E6$, ${D}2D$, and observing the entries on places $(1,1)$,
$(2,2)$, $(3,1)$, and $(3,2)$ in the matrix equation we obtain the equations:

$\displaystyle\lim_{\epsilon\longrightarrow
0}A^{2}_{1,1}=1,\qquad\lim_{\epsilon\longrightarrow
0}A_{1,2}^{2}=0,\qquad\lim_{\epsilon\longrightarrow
0}A_{1,3}A_{1,1}=0,\qquad\lim_{\epsilon\longrightarrow 0}A_{1,2}A_{1,3}=1,$

which obviously can not hold simultaneously.

D1D:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{D}1D\longrightarrow{E}5&&{D}1D\longrightarrow{E}13&&{D}1D\longrightarrow{E}4\\\
\left(\begin{matrix}1&0&0&0\\\ 0&\epsilon&0&1\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\right),&&\left(\begin{matrix}\epsilon^{2}&\epsilon&0&\epsilon^{-1}\\\
\epsilon^{-1}&\epsilon&0&0\\\ 0&0&\epsilon&0\\\
0&0&0&i\epsilon^{-1}\end{matrix}\right),&&\left(\begin{matrix}\epsilon^{2}&\epsilon&0&\epsilon^{-1}\\\
0&\epsilon&1/2&0\\\ 0&0&1&0\\\
0&0&0&i\epsilon^{-1}\end{matrix}\right).\end{array}$

The non-contraction of ${D}1D$ to ${E}6$, ${E}12$, ${S}5$: Demanding that
$\lim\limits_{\epsilon\longrightarrow 0}\widehat{A(\epsilon)}^{\rm
t}B^{16}(1)\widehat{A(\epsilon)}$ converge to the canonical form of one of the
systems ${E}6$, ${E}12$, ${S}5$, and observing the entries on places $(1,1)$,
$(1,2)$, $(4,1)$, and $(4,2)$ in the matrix equation we obtain the equations:

$\displaystyle\lim_{\epsilon\longrightarrow 0}A_{1,1}=\pm
1,\qquad\lim_{\epsilon\longrightarrow
0}A_{1,2}=0,\qquad\lim_{\epsilon\longrightarrow
0}A_{1,2}A_{1,4}=1,\qquad\lim_{\epsilon\longrightarrow 0}A_{1,1}A_{1,4}=0,$

which obviously can not hold simultaneously.

E3:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}}{E}3\longrightarrow{E}5&&{E}3\longrightarrow{E}4\\\
\left(\begin{matrix}1&0&0&0\\\ 0&\epsilon&1&1\\\ 0&0&i&0\\\
0&0&0&1\end{matrix}\right),&&\left(\begin{matrix}\epsilon&0&1&-i\epsilon^{-1}\\\
i\epsilon&\epsilon&i&\epsilon^{-1}\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\right).\end{array}$

D4(b)D:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{D}4(b)D\longrightarrow{E}4&&{D}4(b)D\longrightarrow{E}14&&{D}4(b)D\longrightarrow{E}5\\\
\left(\begin{matrix}\epsilon&0&1&0\\\ 0&\epsilon&0&\epsilon^{-1}\\\
0&0&\epsilon^{2}&0\\\
0&0&0&\frac{1}{\sqrt{2}\epsilon}\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&\epsilon&0&\epsilon^{-1}\\\ 0&0&\epsilon^{2}&0\\\
0&0&0&\frac{1}{\sqrt{2}\epsilon}\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&\epsilon&1&1\\\ 0&0&i&0\\\ 0&0&0&\epsilon\end{matrix}\right),\end{array}$
$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{D}4(b)D\longrightarrow
D1D&&{D}4(b)D\longrightarrow E12&&{D}4(b)D\longrightarrow E13\\\
\left(\begin{matrix}1&0&0&0\\\ 0&-\epsilon&-\epsilon^{-1}&0\\\
0&0&i\epsilon^{-1}&0\\\
0&0&0&\epsilon^{2}\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&-\epsilon&\frac{1}{\sqrt{3}}&-\epsilon^{-1}\\\ 0&0&\sqrt{\frac{2}{3}}&0\\\
0&0&0&\frac{1}{\sqrt{2}\epsilon}\end{matrix}\right),&&\left(\begin{matrix}\epsilon&\epsilon&\frac{1}{2}\epsilon^{-1}&\frac{1}{2}\epsilon^{-1}\\\
i\epsilon&-i\epsilon&-\frac{i}{2}\epsilon^{-1}&\frac{i}{2}\epsilon^{-1}\\\
0&0&\epsilon&0\\\ 0&0&0&\epsilon\end{matrix}\right),\end{array}$
$\displaystyle\begin{array}[]{@{}c@{}}{D}4(b)D\longrightarrow E3\\\
\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&\epsilon\end{matrix}\right).\end{array}$

S3:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{S}3\longrightarrow{E}5&&{S}3\longrightarrow{E}14&&{S}3\longrightarrow{E}4\\\
\left(\begin{matrix}1&0&0&0\\\ 0&\epsilon&i&0\\\ 0&0&1&0\\\
0&0&0&\frac{1}{\sqrt{2}}e^{-i\frac{3\pi}{4}}\end{matrix}\right),&&\left(\begin{matrix}1&0&\epsilon&-\epsilon\\\
0&\epsilon&\epsilon^{3}&\epsilon^{-1}\\\ 0&0&\epsilon^{2}&0\\\
0&0&0&\frac{i}{\epsilon\sqrt{2}}e^{-i\frac{3\pi}{4}}\end{matrix}\right),&&\left(\begin{matrix}\epsilon&0&1&0\\\
0&\epsilon&0&\epsilon^{-1}\\\ 0&0&\epsilon^{2}&0\\\
0&0&0&i\frac{1}{\epsilon\sqrt{2}}e^{-i\pi/4}\end{matrix}\right),\end{array}$
$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{S}3\longrightarrow
D1D&&S3\longrightarrow E13&&S3\longrightarrow E6\\\
\left(\begin{matrix}1&0&0&0\\\ 0&-\epsilon&-\epsilon^{-1}&0\\\
0&0&i\epsilon^{-1}&0\\\
0&0&0&\epsilon^{2}\end{matrix}\right),&&\left(\begin{matrix}\epsilon&\epsilon&\frac{1}{2}\epsilon^{-1}&\frac{1}{2}\epsilon^{-1}\\\
i\epsilon&-i\epsilon&-\frac{i}{2}\epsilon^{-1}&\frac{i}{2}\epsilon^{-1}\\\
0&0&\epsilon&0\\\
0&0&0&\epsilon\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&-\epsilon&-\epsilon^{-1}&-\epsilon^{-1}\\\ 0&0&-i\epsilon^{-1}&0\\\
0&0&0&\frac{e^{i3\pi/4}}{\sqrt{2}\epsilon}\end{matrix}\right),\end{array}$
$\displaystyle\begin{array}[]{@{}c@{}}S3\longrightarrow E3\\\
\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&\epsilon\end{matrix}\right).\end{array}$

S6:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{S}6\longrightarrow{E}12&&{S}6\longrightarrow{E}14&&{S}6\longrightarrow{E}5\\\
\left(\begin{matrix}1&0&0&0\\\ 0&\epsilon&1&\epsilon^{-1}\\\ 0&0&i&0\\\
0&0&0&i\epsilon^{-1}\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&\epsilon&0&\epsilon^{-1}\\\ 0&0&\epsilon^{2}&0\\\
0&0&0&i\epsilon^{-1}\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&\epsilon&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{matrix}\right),\end{array}$
$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{S}6\longrightarrow{E}4&&{S}6\longrightarrow{E}13&&{S}6\longrightarrow{E}18\\\
\left(\begin{matrix}\epsilon^{2}&\epsilon&1&\epsilon^{-1}\\\ 0&\epsilon&0&0\\\
0&0&i&0\\\
0&0&0&i\epsilon^{-1}\end{matrix}\right),&&\left(\begin{matrix}\epsilon&-i\epsilon&\epsilon^{-1}&0\\\
0&\epsilon&i\epsilon^{-1}&\epsilon^{-1}\\\ 0&0&-\epsilon^{-1}&0\\\
0&0&0&i\epsilon^{-1}\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&\epsilon^{-1}&0\\\
0&0&0&\epsilon\end{matrix}\right).\end{array}$

E18:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}{E}18\longrightarrow{E}5&&{E}18\longrightarrow{E}13&&{E}18\longrightarrow{E}4\\\
\left(\begin{matrix}1&0&0&0\\\ 0&\epsilon&0&1\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\right),&&\left(\begin{matrix}\epsilon&\epsilon&\frac{1}{2}\epsilon^{-1}&\frac{1}{2}\epsilon^{-1}\\\
\pm i\epsilon&\mp
i\epsilon&\mp\frac{i}{2}\epsilon^{-1}&\pm\frac{i}{2}\epsilon^{-1}\\\
0&0&\frac{i}{\sqrt{2}}\epsilon^{-1}&0\\\
0&0&0&\frac{i}{\sqrt{2}}\epsilon^{-1}\end{matrix}\right),&&\left(\begin{matrix}\epsilon^{2}&\epsilon&1&(1-i)(2\epsilon)^{-1}\\\
0&\epsilon&0&(1+i)(2\epsilon)^{-1}\\\ 0&0&i&0\\\
0&0&0&(1+i)(2\epsilon)^{-1}\end{matrix}\right).\end{array}$

D3E:

$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}D3{E}\longrightarrow{E}5&&D3{E}\longrightarrow
D1D&&D3{E}\longrightarrow E13\\\ \left(\begin{matrix}1&0&0&0\\\
0&\epsilon&1&1\\\ 0&0&i&0\\\
0&0&0&\epsilon\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&-\epsilon&-\epsilon^{-1}&-1\\\ 0&0&i\epsilon^{-1}&0\\\
0&0&0&i\end{matrix}\right),&&\left(\begin{matrix}\epsilon^{-1}&\epsilon^{2}&\frac{\epsilon}{1-i}&\frac{1}{\epsilon^{2}(1+i)}\\\
i\epsilon^{-1}&\epsilon^{2}&-\frac{\epsilon}{1-i}&\frac{1}{\epsilon^{2}(1-i)}\\\
0&0&-\frac{\sqrt{2}\epsilon}{1+i}&0\\\
0&0&0&\frac{1}{\sqrt{2}\epsilon^{2}}\end{matrix}\right),\end{array}$
$\displaystyle\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}}D3{E}\longrightarrow
E4&&D3{E}\longrightarrow E3&&D3{E}\longrightarrow E18\\\
\left(\begin{matrix}\epsilon&0&1&0\\\ 0&\epsilon&0&\epsilon^{-1}\\\
0&0&\epsilon^{2}&\frac{-1+i}{\sqrt{2}\epsilon}\\\
0&0&0&\frac{1}{\sqrt{2}\epsilon}\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\
0&0&0&\epsilon\end{matrix}\right),&&\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\epsilon&0\\\ 0&0&0&\epsilon^{-1}\end{matrix}\right).\end{array}$

###  5.4 Contractions of the degenerate quadratic algebras
and the lower half of the Askey scheme

Figure 2: Contractions of degenerate systems and the bottom half of the Askey
scheme.

The bottom half of the contraction Askey scheme relating orthogonal
polynomials via contractions to degenerate, singular and free superintegrable
systems is presented in Fig. 2. The top half of the Scheme, relating to
contractions of nondegenerate superintegrable systems can be found in [5] and
the full scheme in [17]. On the left side are the orthogonal polynomials that
realize finite-dimensional representations of the quadratic algebras via
difference or differential operators and on the right those that realize
infinite-dimensional bounded below representations. The arrows from the
nondegenerate superintegrable system $S9$, $E1$, $E8$ and $E3^{\prime}$
correspond to restriction/contractions to degenerate systems, i.e., the
parameters in the 3-parameter potentials are restricted to the case of only 1
parameter and such that one of the symmetry operators becomes a perfect
square. This increases the symmetry algebra of the resulting degenerate
system.

######  Remark 5.7.

For reference, the corresponding potentials are

  * •

$E1$: $V=\alpha(x^{2}+y^{2})+\frac{\beta}{x^{2}}+\frac{\gamma}{y^{2}}$,

  * •

$E3^{\prime}$: $V=\alpha(x^{2}+y^{2})+\beta x+\gamma y$,

  * •

$E8$:
$V=\frac{\alpha(x-iy)}{(x+iy)^{3}}+\frac{\beta}{(x+iy)^{2}}+\gamma\big{(}x^{2}+y^{2}\big{)}$,

on flat space and

  * •

$S9$:
$V=\frac{\alpha}{s_{1}^{2}}+\frac{\beta}{s_{2}^{2}}+\frac{\gamma}{s_{3}^{2}}$,
$s_{1}^{2}+s_{2}^{2}+s_{3}^{2}=1$,

on the 2-sphere.

The arrows from one degenerate superintegrable system to another are the
standard contractions studied above. The singular Laguerre and oscillator
superintegrable systems have singular Hamiltonians, and for these systems
knowledge of the free quadratic algebra does not necessarily determine the
full superintegrable system. The singular Laguerre system is a
restriction/contraction of $E1$. The resulting quadratic algebra is isomorphic
to $\\{H\\}\oplus{\mathfrak{sl}}(2,\mathbb{R})$ [17], in the same sense that
the quadratic algebra of the 2D Kepler system is said to be
${\mathfrak{so}}(3,\mathbb{R})$. This is true only if the system is restricted
to an eigenspace of $H$. The Oscillator system is a contraction of $E6$ and
its quadratic algebra is isomorphic to the 4-dimensional oscillator algebra
[17]. The plane and sphere systems are restriction/contractions of degenerate
superintegrable systems to free superintegrable systems on the plane and the
2-sphere, respectively.

##  6 Conclusions and discussion

This paper is devoted to the study of the geometric quadratic algebras that
correspond to 2D degenerate 2nd order superintegrable systems and general
abstract degenerate quadratic algebras. Since the geometric quadratic algebras
are uniquely determined by their free restrictions we studied only parameter-
free geometric and abstract algebras. The geometric algebras were already
known; in this paper we classified all abstract algebras in Table 3. We
related the geometric and free quadratic algebras in Table 4. We showed that
there were 5 abstract algebras with no geometric counterpart, but that it was
impossible to represent them in phase space.

In Section 3 we derived and classified all Bôcher contractions of 2D
degenerate 2nd order superintegrable systems. In Fig. 2 we showed the
relationship between our results and the bottom half of the Askey scheme. We
derived and classified all abstract contractions of the geometric quadratic
algebras, presenting the results in Table 6. Comparing the Bôcher and abstract
contractions and taking into account the isomorphism of the $E14$ and $S5$
algebras, we see that there is a match except for 6 abstract contractions with
no geometric realization:

$\displaystyle\begin{array}[]{@{}lll}\text{paren algebra}&\text{abstract
contracted algebra}&\\\ \hline\cr S6&D1D&\\\ \hline\cr S3&D1&\\\ \hline\cr
S3&E13\\\ \hline\cr D3E&E18&\\\ \hline\cr E12&E5&\\\ \hline\cr E14&E5&\\\
\hline\cr\end{array}$

In two cases the failure of geometric realization is obvious: It is not
possible to contract a constant curvature space to a Darboux space [7]. This
paper is a partial warm-up for an analogous study of quadratics algebras for
3D superintegrable systems, e.g., [2] and for cubic algebras, e.g., [21].

##  Appendix A Summary of degenerate Laplace and Helmholtz systems

The degenerate superintegrable systems can occur only on the 2-sphere, 2D flat
space, or one of the 4 Darboux spaces. The notation for these systems is taken
from [14, 15]. (We write the systems in classical form; the quantum analogs
have the same potentials and the obvious replacements of classical momenta by
quantum derivatives.) We assume all variables to be complex.

Degenerate complex Euclidean systems $H=p_{x}^{2}+p_{y}^{2}+\alpha V(x,y)$:

  1. 1.

$E18$: $V=\frac{1}{\sqrt{x^{2}+y^{2}}}$, Kepler potential,

  2. 2.

$E3$: $V=x^{2}+y^{2}$, harmonic oscillator,

  3. 3.

$E6$: $V=\frac{1}{x^{2}}$, radial potential,

  4. 4.

$E5$: $V=x$, linear potential,

  5. 5.

$E12$: $V=\frac{x+iy}{\sqrt{(x+iy)^{2}+c^{2}}}$,

  6. 6.

$E14$: $V=\frac{1}{(x+iy)^{2}}$,

  7. 7.

$E4$: $V=x+iy$,

  8. 8.

$E13$: $V=\frac{1}{\sqrt{x+iy}}$.

The last 4 systems are real in Minkowski space.

Degenerate systems on the complex sphere: We use the classical realization for
$o(3,{\mathcal{C}})$ with basis ${J}_{1}=s_{2}p_{s_{3}}-s_{3}p_{s_{2}}$,
${J}_{2}=s_{3}p_{s_{1}}-s_{1}p_{s_{3}}$,
${J}_{3}=s_{1}p_{s_{2}}-s_{2}p_{s_{1}}$, and Hamiltonian
${H}={J}_{1}^{2}+{J}_{2}^{2}+{J}_{3}^{2}+\alpha V$. Here
$s_{1}^{2}+s_{2}^{2}+s_{3}^{2}=1$.

  1. 1.

$S6$: $V=\frac{s_{3}}{\sqrt{s_{1}^{2}+s_{2}^{2}}}$, Kepler analog,

  2. 2.

$S3$: $V=\frac{1}{s_{3}^{2}}$, Higg’s oscillator,

  3. 3.

$S5$: $V=\frac{1}{(s_{1}+is_{2})^{2}}$.

The last system is real on the 2-sheet hyperboloid.

Degenerate systems on Darboux spaces:

  1. 1.

${D1D}$: ${H}=\frac{1}{4x}(p_{x}^{2}+p_{y}^{2})+\frac{\alpha}{x}$,

  2. 2.

$D2D$:
${H}=\frac{x^{2}}{x^{2}+1}(p_{x}^{2}+p_{y}^{2})+\frac{\alpha}{x^{2}+1}$,

  3. 3.

$D3E$:
$H=\frac{1}{2}\frac{e^{2x}}{e^{x}+1}(p_{x}^{2}+p_{y}^{2})+\frac{\alpha}{e^{x}+1}$,

  4. 4.

$D4(b)D$: ${H}=-\frac{\sin^{2}2x}{2\cos
2x+b}(p_{x}^{2}+p_{y}^{2})+\frac{\alpha}{2\cos 2x+b}$.

######  Remark A.1.

Every degenerate system occurs as a “restriction” of at least one
nondegenerate system, although the symmetry algebra grows. For example the
classical nondegenerate $S9$ system has the Hamiltonian

$\displaystyle
H=J_{1}^{2}+J_{2}^{2}+J_{3}^{2}+\frac{a_{1}}{s_{1}^{2}}+\frac{a_{2}}{s_{2}^{2}}+\frac{a_{3}}{s_{3}^{2}}$

and a basis of symmetries

$\displaystyle{L}_{1}={J}_{3}^{2}+a_{1}\frac{s_{2}^{2}}{s_{1}^{2}}+a_{2}\frac{s_{1}^{2}}{s_{2}^{2}},\qquad{L}_{2}={J}_{1}^{2}+a_{2}\frac{s_{3}^{2}}{s_{2}^{2}}+a_{3}\frac{s_{2}^{2}}{s_{3}^{2}},\qquad{L}_{3}={J}_{2}^{2}+a_{3}\frac{s_{1}^{2}}{s_{3}^{2}}+a_{1}\frac{s_{3}^{2}}{s_{1}^{2}},$

where $H={L}_{1}+{L}_{2}+{L}_{3}+a_{1}+a_{2}+a_{3}$. If we let $a_{1}\to 0$,
$a_{2}\to 0$ we obtain the Hamiltonian for $S3$:
$H^{\prime}=J_{1}^{2}+J_{2}^{2}+J_{3}^{2}+\frac{a_{3}}{s_{3}^{2}}$. However,
now

$\displaystyle L_{1}\to L_{1}^{\prime}=J_{3}^{2},\qquad L_{2}\to
L_{2}^{\prime}=J_{1}^{2}+a_{3}\frac{s_{2}^{2}}{s_{3}^{2}},\qquad{L}_{3}\to
L_{3}^{\prime}={J}_{2}^{2}+a_{3}\frac{s_{1}^{2}}{s_{3}^{2}},$

and the restricted system now admits the 1st order symmetry $J_{3}$ as well as
a new 2nd order symmetry $\\{J_{3},L_{2}^{\prime}\\}$. These 4 symmetries are
related by the Casimir. A table with all of the restrictions of nondegenerate
systems on constant curvature spaces to degenerate systems can be found in
[16] and a table with all restrictions of nondegenerate systems on Darboux
spaces can be found in [7].

### Acknowledgments

This work was partially supported by a grant from the Simons Foundation (#
208754 to Willard Miller, Jr and by CONACYT grant (# 250881 to M.A. Escobar).
The author M.A. Escobar is grateful to ICN UNAM for the kind hospitality
during his visit, where a part of the research was done, he was supported in
part by DGAPA grant IN108815 (Mexico). We thank a referee for pointing out the
relevance of references [4, 6, 20].

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