Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates
Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space 𝐸₂ are explored. The study is restricted to Hamiltonians allowing separation of variables 𝑉(𝑥, 𝑦) = 𝑉₁(𝑥) + 𝑉₂(𝑦) in Cartesian coordinates. In particular, the Hamiltonian ℋ admits a polynomial integral of order 𝑁 > 2...
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| Cite this: | Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates. İsmet Yurduşen, Adrián Mauricio Escobar-Ruiz and Irlanda Palma y Meza Montoya. SIGMA 18 (2022), 039, 20 pages |
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| author | Yurduşen, İsmet Escobar-Ruiz, Adrián Mauricio Palma y Meza Montoya, Irlanda |
| author_facet | Yurduşen, İsmet Escobar-Ruiz, Adrián Mauricio Palma y Meza Montoya, Irlanda |
| citation_txt | Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates. İsmet Yurduşen, Adrián Mauricio Escobar-Ruiz and Irlanda Palma y Meza Montoya. SIGMA 18 (2022), 039, 20 pages |
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| description | Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space 𝐸₂ are explored. The study is restricted to Hamiltonians allowing separation of variables 𝑉(𝑥, 𝑦) = 𝑉₁(𝑥) + 𝑉₂(𝑦) in Cartesian coordinates. In particular, the Hamiltonian ℋ admits a polynomial integral of order 𝑁 > 2. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of the formalism are non-linear compatibility conditions and the algebra of the integrals of motion. The case 𝑁 = 5, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case 𝑁 > 2 and a formulation of the inverse problem in superintegrability are briefly discussed as well.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 039, 20 pages
Doubly Exotic Nth-Order Superintegrable Classical
Systems Separating in Cartesian Coordinates
İsmet YURDUŞEN a, Adrián Mauricio ESCOBAR-RUIZ b
and Irlanda PALMA Y MEZA MONTOYA b
a) Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey
E-mail: yurdusen@hacettepe.edu.tr
b) Departamento de F́ısica, Universidad Autónoma Metropolitana-Iztapalapa,
San Rafael Atlixco 186, México, CDMX, 09340 México
E-mail: admau@xanum.uam.mx, cbi2153013099@izt.uam.mx
Received December 18, 2021, in final form May 16, 2022; Published online May 27, 2022
https://doi.org/10.3842/SIGMA.2022.039
Abstract. Superintegrable classical Hamiltonian systems in two-dimensional Euclidean
space E2 are explored. The study is restricted to Hamiltonians allowing separation of vari-
ables V (x, y) = V1(x) + V2(y) in Cartesian coordinates. In particular, the Hamiltonian H
admits a polynomial integral of order N > 2. Only doubly exotic potentials are considered.
These are potentials where none of their separated parts obey any linear ordinary differential
equation. An improved procedure to calculate these higher-order superintegrable systems
is described in detail. The two basic building blocks of the formalism are non-linear com-
patibility conditions and the algebra of the integrals of motion. The case N = 5, where
doubly exotic confining potentials appear for the first time, is completely solved to illustrate
the present approach. The general case N > 2 and a formulation of inverse problem in
superintegrability are briefly discussed as well.
Key words: integrability in classical mechanics; higher-order superintegrability; separation
of variables; exotic potentials
2020 Mathematics Subject Classification: 70H06; 70H33; 70H50
1 Introduction
For a classical Hamiltonian system with n degrees of freedom, the existence of n integrals of
motion in involution is required to make it integrable in the Liouville sense. These integrals
must be well-defined functions in the phase space and functionally independent. On the other
hand, a superintegrable system possesses k additional integrals of motion being k = n − 1 the
maximum possible number. The concept of superintegrability can be defined both in classical
and quantum mechanics and it has been studied extensively for a very long period of time.
The outcome of such a long period of research activity has far reaching consequences both in
mathematical and physical points of view. There exist several exhaustive review articles in
literature which describe the history and current status of this topic [34, 51].
Starting with a spherically symmetric standard Hamiltonian (i.e., the potential V = V (r)
being velocity and spin independent), there exist only two superintegrable systems, namely
the Kepler–Coulomb and the harmonic oscillator. Actually, these two potentials are exactly
the ones which appear in the celebrated Bertrand’s theorem [3, 21]. Superintegrability of the
Kepler–Coulomb problem is due to the existence of the conserved Laplace–Runge–Lenz vector
[21, 40, 41, 60] whilst in the case of the harmonic oscillator is a consequence of the existence of
the quadrupole Jauch–Fradkin tensor [19, 30].
mailto:yurdusen@hacettepe.edu.tr
admau@xanum.uam.mx
cbi2153013099@izt.uam.mx
https://doi.org/10.3842/SIGMA.2022.039
2 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
The systematic investigation of superintegrability has been initiated by Pavel Winternitz and
his collaborators in 1965 [20]. They first considered quadratic superintegrability in Euclidean
spaces and the subject has been subsequently developed into many directions by several authors
since then. For example, its close relation with multiseparability was studied in detail in the
references [17, 18, 20, 31, 42, 49], the search for superintegrable systems in 2- and 3-dimensional
spaces of constant and nonconstant curvature has been carried out in the works [25, 26, 27, 31,
32, 33, 35, 36, 37, 50] and their generalizations to n-dimensions have been analyzed in the papers
[38, 39, 59].
Another important research direction in this field is to consider the Hamiltonians with mag-
netic field and/or spin. Superintegrability with magnetic field was first explored in the articles
[5, 10] and much recently developed in the articles [4, 43, 44]. The systematic investigation of
integrability and superintegrability for systems involving particles with spin was initiated in the
reference [68] and subsequently all the rotationally invariant superintegrable systems in E3 were
classified in the articles [9, 69, 70]. On the other hand, spin dependent superintegrable systems
were studied in the works [53, 54] for matrix potentials simulating charged or neutral fermions
with non-trivial dipole moment in the presence of an electric field.
Still another interesting direction is to go beyond quadratic superintegrability, i.e., the gene-
ral theory of higher-order superintegrability. Initial pioneering works were the articles of Drach
[11, 12], where 10 potentials allowing third-order integrals of motion were announced. However,
much later it was shown that 7 of these potentials are actually reducible, the third-order integral
is the Poisson commutator of two second-order integrals [58, 65]. Once again, the systematic
investigation of higher-order superintegrability, in particular the third-order one has been initi-
ated by Pavel Winternitz and his collaborators in the articles [22, 23, 45, 48, 55, 64]. Almost
around the same time higher-order symmetry operators were calculated for the Schrödinger ope-
rator and the determining equations for the corresponding integrals of motion appeared in [52].
Nevertheless, it was soon realized that the analysis became very complicated and some new ways
of approaching to the problem of higher-order superintegrability have to be considered.
After the publication of the seminal paper “An infinite family of solvable and integrable
quantum systems on a plane” [62], the direction of the research has been thoroughly shifted
to higher-order integrability/superintegrability [56, 57, 63]. Moreover, in order to make them
more easily tractable, new techniques and methods have been implemented in the study of
higher-order integrable and superintegrable systems [6, 7].
From our point of view one of the main issues on higher-order superintegrability is the clas-
sification of the superintegrable potentials. In the case of 2D separable potentials in Cartesian
coordinates, an Nth-order superintegrable system appeared for the first time in [61], where the
existence of nonlinear equations for the potential which makes the general problem much more
complicated was stressed as well. Recently in 2018, by means of a systematic study an infinite
2-parametric family of superintegrable potentials embracing those found in [28, 61] was pre-
sented in the paper [24] by Grigoriev and Tsiganov. Their key element to construct polynomial
integrals of motion is the addition theorems for the action-angle variables, especially the Cheby-
shev theorem applied to integrals on differential binomials (see also [66, 67]). Such an elegant
approach has the advantage that it uses the action-angle variables which play a fundamental
role in classical mechanics.
However, unlike the present direct approach some systems can be missed and the gene-
ralization to the quantum case is not straightforward. The explicit list of all Nth-order 2D
(polynomial) superintegrable potential separating in Cartesian coordinates is far to be complete.
Throughout many recent research activity in the field of higher order superintegrability, it has
been clarified that three types of potentials can occur, namely the standard, the doubly exotic
and singly exotic potentials. Standard ones are solutions of a linear compatibility condition
for the determining equations that govern the existence of a higher-order polynomial integral
Doubly Exotic Nth-Order Superintegrable Classical Systems Separating 3
of motion. For doubly exotic potentials this linear compatibility condition is satisfied trivially,
it is identically zero, and the potentials satisfy non-linear equations. These classes of potentials,
appearing in classical and quantum superintegrable systems have been studied both in Cartesian
and polar coordinates [1, 14, 15, 16, 47].
The aim of this work is to establish in detail general properties of Nth-order superintegrable
classical systems that allow separation of variables in Cartesian coordinates. It can be considered
as the classical counterpart of the general study on quantum superintegrable systems treated
in [13]. However, unlike the latter, in this work we also study the algebra of the integrals of
motion and provide exhaustive results for the case N = 5. In particular, it contains the classical
analogues of all the quantum doubly exotic potentials obtained in [1] explicitly. We emphasize
that for doubly exotic potentials, unlike the doubly standard ones, the limit from the quantum
to the corresponding classical system (i.e., ℏ → 0) is singular for all the cases studied in the
present work. Thus, the corresponding quantum and classical solutions are not connected at all.
The Painlevé property characterizing the relevant determining equations in the quantum sys-
tems is completely lost in the classical case. In addition, a formulation of inverse problem in
superintegrability is briefly discussed as well.
In the present article we focus on 2D classical Hamiltonian systems that are separable in
Cartesian variables (x, y) and they also admit an extra polynomial integral of order N > 2. The
generic Hamiltonian is given by
H = H1(x) +H2(y) ≡
1
2
(
p21 + p22
)
+ V1(x) + V2(y), (1.1)
where pi, i = 1, 2, are the canonical momenta conjugate to x and y, respectively. It describes
a two-dimensional particle with unit massm = 1 moving in the potential V (x, y) = V1(x)+V2(y).
Thus, the phase space is four-dimensional. These systems are trivially second-order integrable
because in addition to the Hamiltonian (1.1) they admit, for any V1(x) and V2(y), another
2nd-order symmetry of the form
X = H1(x)−H2(y) =
1
2
(
p21 − p22
)
+ V1(x)− V2(y), (1.2)
which Poisson commutes (i.e., {H,X}PB = 0) with the Hamiltonian (1.1). The existence of
an Nth-order third integral Y, makes the system Nth-order superintegrable (more integrals of
motion than degrees of freedom). In this case, the system is maximally superintegrable. Notice
that H (1.1) is S2-invariant under the permutation x ⇔ y whilst the integral X is anti-invariant.
From a physical point of view we are looking for 2D potentials V (x, y) = V1(x) + V2(y) for
which all the bounded trajectories are closed and periodic. It is worth mentioning that (1.1) can
also be interpreted as the Hamiltonian of the relative motion of a two-body problem on the plane
with translational invariance. In this case, (x, y) are nothing but the Cartesian coordinates of
the relative vector r = r1 − r2 ≡ (x, y) between the two bodies.
The outline of the paper is as follows. In Section 2, for an arbitrary potential V (x, y) not
necessarily separable in a coordinate system we revisited the so called determining equations
governing the existence of a general Nth-order polynomial integral of motion YN . In particular,
the dominant Nth-order terms in YN lie in the enveloping algebra of the Euclidean Lie alge-
bra e(2). From the next leading terms in YN , a linear compatibility condition (LCC) can be
obtained for the potential V only. The case of a separable potential in Cartesian coordinates is
then analyzed in Sections 3–4, where we show and describe a well of determining equations and
derive the first non-linear compatibility condition for the potential alone. The general form of
the potentials is determined by solving these compatibility conditions. Afterwards, the surviv-
ing determining equations become linear and can be solved. In Section 5, based on the LCC,
we introduce the doubly exotic potentials. A general formula for the corresponding integral YN
4 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
V (x, y) = V1(x) + V2(y)
m
y
x
Figure 1. The Hamiltonian (1.1) describes a particle with unit mass m = 1 moving in a two-dimensional
potential V (x, y) = V1(x) + V2(x).
is given. In Section 6 we discuss the role of the algebra of the integrals of motion in the search
of superintegrable potentials, and a formulation of inverse problem in superintegrability is com-
mented. Section 7 is devoted to the known examples with N = 3, 4. Finally, in Sections 8
and 9 we consider the case N = 5 and derive in detail all possible doubly exotic potentials. For
conclusions see Section 10.
2 Superintegrability: existence of an Nth-order polynomial
integral
In the present article we are considering Hamiltonian systems separable in Cartesian coordi-
nates and hence they are second order integrable by construction. To further search for the
superintegrability, we need to give the conditions for the existence of an additional integral of
motion, which is a polynomial of order N > 2 in variables p1, p2. Although the general ideas
for the existence of a Nth-order integral is given in [29, 57], here we would like to summarize
those results for the sake of completeness.
2.1 General form
The most general form of an Nth-order polynomial integral YN is given by
YN =
[N
2
]∑
ℓ=0
N−2ℓ∑
j=0
fj,2ℓ p
j
1p
N−j−2ℓ
2 , (2.1)
see [29, 57], where fj,2ℓ = fj,2ℓ(x, y, V ) are assumed to be real functions which depend on the
coordinates x and y and the potential V (x, y).
The integral YN (2.1) can be conveniently rewritten as follows
YN = WN + lower order terms, (2.2)
where the leading term WN in (2.2)
WN =
∑
0≤m+n≤N
AN−m−n,m,nL
N−m−n
z pm1 pn2 , (2.3)
plays a fundamental role since it governs the existence or non-existence of the integral YN , here
AN−m−n,m,n are (N+1)(N+2)
2 real parameters and Lz = xp2 − yp1 is the z-component of angular
Doubly Exotic Nth-Order Superintegrable Classical Systems Separating 5
momentum. If the quantity YN Poisson commutes with the Hamiltonian (1.1) then the system
becomes Nth-order (N > 1) superintegrable. In fact, for 2D systems it would correspond to
maximal superintegrability.
2.2 The determining equations
The Poisson bracket of YN (2.1) with the Hamiltonian H (1.1) gives a polynomial, in p1 and p2,
of degree (N + 1). Explicitly, we have
{H,YN}PB =
N+1∑
n1+n2=0
Mn1,n2p
n1
1 pn2
2 ,
where the coefficients Mn1,n2 = Mn1,n2(x, y; fj,2ℓ, V,N) depend on the variables x, y, the func-
tions fj,2ℓ appearing in the integral YN , the potential V (x, y) we are looking for, and they also
carry an N -dependence. Superintegrability requires
Mn1,n2 = 0, n1 + n2 = 0, 1, 2, . . . , (N + 1), (2.4)
({H,YN} = 0). For an arbitrary potential V (x, y) not necessarily separable, the system (2.4) is
equivalent to the following set of determining equations (DE):
(∂xfj−1,2ℓ + ∂yfj,2ℓ)−
[
(j + 1)fj+1,2ℓ−2
]
∂xV −
[
(N − 2ℓ+ 2− j)fj,2ℓ−2
]
∂yV = 0, (2.5)
ℓ = 0, 1, 2, . . . ,
[
N
2
]
, j = 0, 1, 2, . . . , (N − 2ℓ). In (2.5), the real functions fj,ℓ ≡ 0 identically for
ℓ < 0 and j < 0 as well as for j > N − 2ℓ (further details can be found in [29, 57]). The DE
correspond to the vanishing of all the coefficients, in the Poisson bracket {H,YN}PB, multiplying
the momentum terms of order n1+n2 = k = N+1, N−1, N−3, . . . , (N+1−2ℓ). In particular,
for odd N the coefficient multiplying the zero order term is simply f1,N−1V
′
1 + f0,N−1V
′
2 = 0,
obtained from (2.5) by making the replacement ℓ → ℓ + 1. The DE govern the existence of
the integral YN . In general, the system (2.5) is overdetermined. If the potential V (x, y) is not
known a priori, then it must be calculated from the compatibility conditions of the DE.
The structure of the DE (2.5) can be summarized as follows:
� The set of DE (2.5) can be seen as a well of recursive equations. The coefficients fj,2ℓ
in YN depend on the preceding fj,2k, 0 ≤ k < ℓ.
� The bottom level of equations (2.5) corresponds to ℓ = 0. The associated DE do not depend
on V , thus, allowing exact solvability. Indeed, they define the coefficient-functions fj,0,
j = 0, 1, 2, . . . , N . The explicit expression for fj,0 is given by
fj,0 =
N−j∑
n=0
j∑
m=0
(
N − n−m
j −m
)
AN−n−m,m,nx
N−j−n(−y)j−m,
see [29, 57]. Accordingly, the leading part (2.3) of YN is a polynomial of order N in the
enveloping algebra of the Euclidean Lie algebra e(2) with basis {p1, p2, Lz}.
� The 2nd level of DE (2.5) occurs at ℓ = 1. They provide a linear compatibility condition
(LCC) for the potential V only. For arbitrary potential, this linear PDE can be written
in the compact form [29, 57]
N−1∑
j=0
(−1)j∂N−1−j
x ∂j
y
[
(j + 1)fj+1,0∂xV + (N − j)fj,0∂yV
]
= 0. (2.6)
This above equation is a necessary but not sufficient condition for {H,YN} = 0. Also, in
the quantum case the LCC remains identical to (2.6). However, the corresponding DE do
acquire ℏ-dependent terms.
6 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
� Beginning from ℓ = 2, the DE (2.5) will lead to nonlinear compatibility conditions (NLCC)
for the potential V alone. We should remind here that in the quantum case these NLCC,
unlike the LCC, do depend non-trivially on ℏ. Hence, the classical and quantum cases can
greatly differ, and it requires to treat them separately.
3 Superintegrable potentials separable in Cartesian coordinates
3.1 The linear compatibility condition
In the case of a separable potential the LCC (2.6) leads to the ordinary differential equations
for V1(x)
N−1∑
j=0
(j + 1)!
N−j−1∑
n=0
(
N − 1− n
j
)
AN−1−n,1,n
(
d
dx
)N−j+1[
xN−j−n−1V ′
1(x)
]
= 0, (3.1)
and
N−1∑
j=0
(j+ 1)(j+ 1)!(−1)2j+1
N−j−1∑
n=0
(
N− n
j+ 1
)
AN−n,0,n
(
d
dx
)N−j+1[
xN−j−n−1V ′
1(x)
]
= 0. (3.2)
For superintegrability, {H,YN} = 0, these two linear equations (3.1) and (3.2) must be simul-
taneously satisfied. Similarly, for V2(y) there exist two ODEs which can be obtained from (3.1)
and (3.2) using the symmetry x ↔ y, respectively.
4 The first nonlinear compatibility condition
In the case of an arbitrary odd N ≥ 3 polynomial integral of motion YN , following the derivation
presented in [13] we describe the procedure to construct the first NLCC in detail. In general,
this equation obtained from the DE (2.5) with ℓ = 2 provides the form of the doubly exotic
potentials, see below.
As a first step, one solve the DE (2.5) with ℓ = 1. These equations define all the coefficient-
functions fj,2 appearing in the integral (2.1).
Secondly, from the DE (2.5) with ℓ = 2 we compute the (N − 3) functions fj,4 except those
with j = N−5
2 and j = N−3
2 . Eventually, we arrive at the equations
∂yfN−5
2
,4 = F̃N−5
2
, ∂xfN−5
2
,4 + ∂yfN−3
2
,4 = F̃N−3
2
, ∂xfN−3
2
,4 = F̃N−1
2
, (4.1)
here the F̃ ’s, by construction, are real functions that solely depend on the potential V (and its
derivatives).
Finally, from (4.1) it follows the equation
∂2
xF̃N−5
2
+ ∂2
y F̃N−1
2
− ∂2
x,yF̃N−3
2
≡ 0,
which gives the aforementioned NLCC for the potential V . In the case of arbitrary even N ≥ 4,
the steps are quite similar, see details in [13].
From (2.5), it follows that more NLCC occur for each value of ℓ = 3, 4, . . . ,
[
N
2
]
. Nevertheless,
these additional equations will simply restrict the general solution of the potential V found from
the previous NLCC with ℓ = 2.
Therefore, for a separable potential V = V1(x) + V2(y) the set of DE with ℓ = 0 are given
by a system of ODEs which do not depend on V and they specify the coefficient-functions fj,0
Doubly Exotic Nth-Order Superintegrable Classical Systems Separating 7
(j = 0, 1, 2, . . . , N) appearing in (2.1). Then, the next level of DE with ℓ = 1 provide a LCC
for the potential alone and they also determine the functions fj,2 (j = 0, 1, 2, . . . , N − 2). At all
further levels ℓ ⩾ 2 the DE and their compatibility conditions are nonlinear ODEs for V . These
compatibility conditions are instrumental to specify the general form of the potential V .
5 Doubly exotic potentials
Hereafter, we will restrict ourselves to the case of doubly exotic potentials. These potentials
satisfy the LCC (2.6) trivially. In particular, the two linear ODEs (3.1) and (3.2) vanish iden-
tically for any V1(x). Hence, this LCC does not impose any constraint for V1(x) nor for V2(y).
This situation occurs when the number of coefficients AN−m−n,m,n that figure in the LCC is
less that those appearing in the integral YN (2.1). In this case, we simply put equal to zero the
coefficients AN−m−n,m,n in the LCC, thus, it vanishes identically, but still the integral YN is of
order N . In general, based on the LCC (2.6) one can classify the Nth-order superintegrable
systems into three major classes: doubly exotic potentials, singly exotic potentials and standard
potentials (see [13]). This general classification is summarized in Table 1.
Table 1. Classification of Nth-order superintegrable classical systems (N > 2) separating in Cartesian
coordinates. For a fixed value of N , there exist three generic types of potentials: doubly standard, doubly
exotic and singly exotic potentials.
Potential Doubly standart Doubly exotic Singly exotic
V = V1(x) + V2(y) potentials potentials potentials
Classical Both functions Both V1(x), V2(y) The x-component
superintegrable V1(x), V2(y) satisfy obey a NLCC, V1(x) satisfies
systems non-trivially a non-linear ODE a linear/non-linear
the LCC, which do not pass ODE whilst y-component
a linear ODE the Painlevé test. V2(y) obeys
The LCC is identically zero a non-linear/linear OD
From this point of view, Cases 1–3 of Proposition 1 presented in [24] are doubly exotic
potentials for n1, n2 > 1 whilst Cases 4–5 at n > 1 can not be doubly standard ones. Moreover,
the aforementioned Cases 1–3 are nothing but particular solutions of the present direct approach.
It is worth mentioning that we solely consider potentials where neither the x-part V1(x) nor the
y-part V2(y) are constant functions.
5.1 Integral YN for doubly exotic potentials
In the present work we will focus on doubly exotic potentials. In this case, the corresponding
Nth-order terms of the integral YN (2.1) are given by
WN = A0,N,0p
N
1 +A0,0,NpN2 +AN−4,2,2L
N−4
z p21p
2
2
+
∑
4<m+n<N ; |m−n|<N−4
AN−m−n,m,nL
N−m−n
z pm1 pn2
+
∑
0≤m+n=N ; |m−n|≤N−4
A0,m,np
m
1 pn2 . (5.1)
Therefore, WN (5.1) carries 1
2
(
10 − 5N + N2
)
less constants AN−m−n,m,n than the generic
term (2.1). For large N , the number of constants AN−m−n,m,n in WN grows quadratically
with N . Notice that Lz occurs in WN starting from N = 5. For N ≥ 5, it can also contain
terms like Lz, L
2
z, L
3
z, . . . , L
N−4
z only.
8 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
Let us give the most general (leading) term WN of the integral YN,doubly exotic for N = 3, 4, 5
explicitly
W3 = A030p
3
1 +A003p
3
2,
W4 = A040p
4
1 +A004p
4
2 +A022p
2
1p
2
2,
W5 = A050p
5
1 +A005p
5
2 +A032p
3
1p
2
2 +A023p
2
1p
3
2 +A122Lzp
2
1p
2
2.
6 ODEs versus algebraic equations. Algebras of integrals
of motion
In the search of Nth-order superintegrable potentials one faces the problem of solving an overde-
termined system of ODEs where some of them are non-linear. Moreover, the number of involved
equations increases with N . Therefore, the direct approach of solving all the DE (2.5) is far
from being an efficient method. In order to simplify it, in the present consideration we propose
to combine two basic elements, namely the non-linear compatibility conditions and the use of
the algebra of the integrals (see below). As a result, in some cases the ODEs are reduced to
pure algebraic equations.
From X and YN , we introduce the quantity
C ≡ {YN ,X}PB, (6.1)
which is a polynomial function in p1 and p2 of degree (N+1). If YN is an integral of motion, then
by construction C (6.1) is also conserved. The closure of the algebra generated by the integrals
of motion (H,X ,YN , C) is guaranteed by the property of maximal superintegrability. The main
question we aim to explore is the appearance and utility of a closed polynomial algebra.
It is important to mention that the study of the algebraic structure of the integrals of motion
has been proven to be fruitful in the classification of higher-order superintegrable classical and
quantum systems [8, 46].
Also, the explicit results obtained in Section 9 suggests to explore the inverse problem, namely
we take two polynomial functions A and B in momentum variables (p1, p2) and construct the
new object C = {A,B}PB. Now, let us assume that the algebra generated by (A,B, C) is a closed
polynomial algebra with polynomial coefficients in H. The question is under what conditions
these closure relations imply that A and B are integrals, i.e., they Poisson commute with H?
7 Lowest order cases N = 3 and N = 4:
doubly exotic potentials
7.1 Case N = 3
The general integral (2.1) at N = 3 is given by
Y3 = f3,0p
3
1 + f0,0p
3
2 + f1,0p1p
2
2 + f2,0p2p
2
1 + f1,2p1 + f0,2p2. (7.1)
The first set of DE (2.5) with ℓ = 0 corresponds to the vanishing of all the coefficients, in the
Poisson bracket {H,Y3}PB, multiplying the highest momentum terms of order 4. They can be
solved directly to give the functions fj,0. For doubly exotic potentials, they read
f30 = A030, f20 = 0, f10 = 0, f00 = A003.
Thus, (7.1) reduces to
Y3 = A030p
3
1 +A003p
3
2 + f1,2p1 + f0,2p2. (7.2)
Doubly Exotic Nth-Order Superintegrable Classical Systems Separating 9
The next set of DE is obtained by setting ℓ = 1 in (2.5). They correspond to the vanishing of all
the coefficients, in the Poisson bracket {H,Y3}PB, multiplying the (next-to-leading) momentum
terms of order 2. These DE take the form
f
(1,0)
1,2 = 3A030V
′
1 , f
(0,1)
1,2 + f
(1,0)
0,2 = 0, f
(0,1)
0,2 = 3A003V
′
2 . (7.3)
The compatibility condition of the above system (7.3) does not provide further information on
the potentials functions. However, the first and third equations can be solved immediately, they
define the functions f0,2 and f1,2
f1,2 = 3A030V1 + u2(y), f0,2 = 3A003V2 + u1(x), (7.4)
where u1(x) and u2(y) are arbitrary functions of x and y, respectively. Substituting (7.4) into
the second equation in (7.3) we obtain the equation u′1 + u′2 = 0. Therefore,
u1 = α1 + βx, u2 = α2 − βy,
here α1, α2 are constants of integrations whilst β is a separation constant. Finally, the last
determining equation corresponds to the vanishing of the coefficient, in the Poisson bracket
{H,Y3}PB, of order zero in momentum variables. Explicitly, it takes the form
3A030V1V
′
1 + 3A003V2V
′
2 + α2V
′
1 − βyV ′
1 + βxV ′
2 + α1V
′
2 = 0. (7.5)
Non trivial solutions of (7.5) correspond to separation of variables, namely β = 0. In this case,
(7.5) leads to the following uncoupled equations
3A030V1V
′
1 + α2V
′
1 = λ, 3A003V2V
′
2 + α1V
′
2 = −λ,
being λ ̸= 0 (otherwise the functions V1,2 are just constants) the corresponding separation
constant. Eventually, we arrive to the solutions
V1 =
√
2λ
3A030
√
x, V2 =
√
−2λ
3A003
√
y. (7.6)
In general, the Poisson bracket C = {Y3,X}PB between YN (2.1) with N = 3 and X (1.2) is
a polynomial in the variables p1 and p2 of degree four. However, for the potential functions (7.6),
we obtain C ∝ λ. Hence, in this case the algebra of the three integrals of motion (C,Y3,X )
takes the form
{C,X}PB = 0, {C,Y3}PB = 0.
It is worth mentioning that the family of superintegrable potentials with
YN = A0N0p
N
1 +A00NpN2 + lower order terms,
has been analyzed in [28] by means of Heisenberg-type higher order symmetries. For this family
all three conserved quantities (H,X ,YN ) admit separation of variables in Cartesian coordinates.
However, such an approach does not allow us to obtain all the doubly exotic potentials with
non-separable integrals YN .
10 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
V = V1(x) + V2(y)
0
0.5
1.0
1.5
2.0
0
0.2 0.4
x
0.6
0.8
1.0
0
0.2
0.4 y
0.6
0.8
1.0
Figure 2. The doubly exotic potential (7.6) corresponding to N = 3. It admits the third-order integ-
ral Y3 (7.2). The values A030 = −A003 = 2λ
3 were used.
7.2 Case N = 4
In this case N = 4, for a doubly exotic potential the most general expression of the fourth-order
integral Y4 reads
Y4 = A040p
4
1 +A004p
4
2 +A022p
2
1p
2
2 + lower order terms,
where A040, A004 and A022 are real constants. It can immediately be rewritten as follows
Y4 = A040(H+ X )2 +A004(H−X )2 +A022(H+ X )(H−X ) + lower order terms. (7.7)
Now, without losing generality, one can always add to (7.7) any arbitrary function of the second
order trivial integrals H and X . This implies that no bona fide doubly exotic potentials, with
a non-trivial fourth order integral, exist.
8 Case N = 5: doubly exotic potentials
We can write the most general 5th-order polynomial integral Y5 in the form
Y5 =
2∑
ℓ=0
5−2ℓ∑
j=0
fj,2ℓ p
j
1p
5−j−2ℓ
2 . (8.1)
8.1 Determining equations
Putting ℓ = 0 in (2.5) corresponds to the vanishing of all the coefficients, in the Poisson bracket
{H,Y5}PB, multiplying the highest momentum terms of order 6. They can be solved directly to
give the functions fj,0
f50 = A050, f40 = 0, f30 = A032 − yA122,
f20 = A023 + xA122, f10 = 0, f00 = A005,
where the condition that the LCC (2.6) is satisfied trivially was imposed, namely we consider
doubly exotic potentials only. It implies that the existence or non-existence of fifth order dou-
blyexotic potentials is governed by 5 parameters A050, A005, A032, A023, A122 only. The next set
of DE are obtained by setting ℓ = 1:
f0,2
(0,1) = 5A005V
′
2 ,
Doubly Exotic Nth-Order Superintegrable Classical Systems Separating 11
f1,2
(0,1) + f0,2
(1,0) = 2(A023 + xA122)V
′
1 ,
f1,2
(1,0) + f2,2
(0,1) = 3(A032 − yA122)V
′
1 + 3(A023 + xA122)V
′
2 ,
f3,2
(0,1) + f2,2
(1,0) = 2(A032 − yA122)V
′
2 ,
f3,2
(1,0) = 5A050V
′
1 . (8.2)
Now, the three DE (2.5) with ℓ = 2 are given by
f1,4
(1,0) = 3f3,2V
′
1 + f2,2V
′
2 ,
f1,4
(0,1) + f0,4
(1,0) = 2(f2,2V
′
1 + f1,2V
′
2),
f0,4
(0,1) = 3f0,2V
′
2 + f1,2V
′
1 . (8.3)
Next, following the discussion of Section 4 we obtain from (8.2) the functions f3,2, f2,2, f1,2, f0,2
in terms of V (see below). Afterwards, the r.h.s. in (8.3) would depend (non-linearly) on V and
its derivatives alone. Consequently, (8.3) leads to the first NLCC in the form
∂2
xf
(0,1)
0,4 + ∂2
yf
(1,0)
1,4 − ∂x∂y
(
f
(1,0)
0,4 − f
(0,1)
1,4
)
= 0. (8.4)
Finally, the last determining equation with ℓ = 2 reads
f1,4V
′
1 + f0,4V
′
2 = 0.
8.2 The (first) NLCC
The DE with ℓ = 1 (8.2) define the four functions f0,2, f1,2, f2,2 and f3,2 appearing in the
integral Y5 (8.1) in front of the cubic terms (pi1p
j
2 with i+ j = 3). Explicitly
f0,2 = 2xA122T
′
1(x) + 2A023T
′
1(x) +A122T1(x) + 5A005T
′
2(y) + α1 − β4x
3 + σ3x
2 + α2x,
f1,2 = y(−3A122T
′
1(x)− α2 + 3β4x
2 − 2σ3x) + 3A032T
′
1(x) + ν1 + ν3x
2 − σ2x,
f2,2 = x(3A122T
′
2(y)− β2 − 3β4y
2 − 2ν3y) + 3A023T
′
2(y) + σ1 + σ3y
2 + σ2y,
f3,2 = −2yA122T
′
2(y) + 2A032T
′
2(y)−A122T2(y) + 5A050T
′
1(x)
+ β1 + β4y
3 + ν3y
2 + β2y, (8.5)
where
V1(x) ≡ T ′
1(x), V2(y) ≡ T ′
2(y).
Next, substituting (8.3) and (8.5) into (8.4) we obtain the following non-linear compatibility
condition (NLCC)
NLCC = T1
(4)(3T ′
1(A032 − yA122) + ν1 + 3β4x
2y + x(−σ2 + ν3x− 2σ3y)− α2y)
+ T2
(4)(3T ′
2(xA122 +A023) + σ1 − 3β4xy
2 + y(σ2 + σ3y − 2ν3x)− β2x)
+ T1
(3)(−9yA122T
′′
1 + 9A032T
′′
1 − 4σ2 + 8ν3x+ 24β4xy − 8σ3y)
+ T2
(3)(9xA122T
′′
2 + 9A023T
′′
2 + 4σ2 − 8ν3x− 24β4xy + 8σ3y)
+ 12ν3T
′′
1 + 12σ3T
′′
2 − 36β4xT
′′
2 + 36β4yT
′′
1 = 0, (8.6)
where α1, β
′s, ν ′s and σ′s are constants to be determined. We have the freedom to replace T1(2)
by T1(2)+c for some real constant c to simplify the expressions. Also we can shift the variables x
or y. Notice that the constants A050 and A005 do not appear in (8.6).
In terms of the parameters A5−m−n,m,n that define the existence or non-existence of the
integral Y5, we identify two cases for which the above NLCC (8.6) admits separation of variables
in Cartesian coordinates, namely:
12 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
(i) A122 ̸= 0, A023 = A032 = 0,
(ii) A2
023 +A2
032 ̸= 0, A122 = 0,
with A050 and A005 arbitrary. These two cases are S2-invariant under the permutation x ⇔ y
(thus, p1 ⇔ p2). Let us recall that the Hamiltonian H and the integral X are S2-invariant and
S2-antiinvariant, respectively. Moreover, if
(iii) A122 = A023 = A032 = 0,
with A2
050 + A2
005 ̸= 0, the NLCC degenerates into a linear equation which must be identically
zero for doubly exotic potentials. In such a case the NLCC does not provide any information on
the potential. As a result of calculations, the cases (i), (ii) and (iii) are the only generic ones
that satisfy all the DE.
9 Results
9.1 Superintegrable potentials
Below, adopting the notation introduced in [1] we present the full list of doubly exotic fifth-order
(N = 5) superintegrable potentials:
Case (i).
• The system Q1: A122 ̸= 0, A032 = A023 = A050 = A005 = 0. This system corresponds
to A122 = 1, all other parameters Aijk = 0. In this case, by solving all the DE (8.2)–(8.4) we
eventually arrive to the first-order nonlinear ODEs
(T ′
1)
2 − 2β4x
2T ′
1 − 4β4T1x+ β2
4x
4 = 0,
(T ′
2)
2 − 2β4y
2T ′
2 − 4β4T2y + β2
4y
4 = 0, (9.1)
β4 ̸= 0 is a real constant. The corresponding fifth-order integral of motion is given by
Y(Q1)
5 = p21p
3
2x− p31p
2
2y + p31
(
−2yT ′
2 − T2 + β4y
3
)
+ p32
(
2xT ′
1 + T1 − β4x
3
)
+ p21p2x
(
3T ′
2 − 3β4y
2
)
+ p1p
2
2y
(
3β4x
2 − 3T ′
1
)
+ p1
(
−3
2
β4x
2y2T ′′
2 + 3β4y
3T ′
1 +
3
2
x2T ′
2T
′′
2 − 6yT ′
1T
′
2 − 3T2T
′
1
)
+ p2
(
3
2
β4x
2y2T ′′
1 − 3β4x
3T ′
2 −
3
2
y2T ′
1T
′′
1 + 6xT ′
1T
′
2 + 3T1T
′
2
)
. (9.2)
From X and Y(Q1)
5 , we built the quantity
C ≡
{
Y(Q1)
5 ,X
}
PB
,
which is a polynomial function in p1 and p2 of sixth degree. By construction, it is an integral
when (9.1) are satisfied. Now, if we demand that the three elements
(
X , Y(Q1)
5 , C
)
generate
a closed polynomial algebra we eventually arrive to a nonlinear first-order differential equation
for T1(x) and similarly for T2(y). Therefore, from these equations and (9.1) we can eliminate
the first-derivative T ′
1, T
′
2 terms and obtain an algebraic equation for both T1(x) and T2(y). The
solutions of such algebraic equations turn out to be the general solutions of (9.1). Explicitly,
these algebraic equations take the form
3β4T
2
1 + 8β
3/2
4 x3/2T
3/2
1 + 6β2
4x
3T1 − β3
4x
6 − δ = 0,
3β4T
2
2 + 8β
3/2
4 y3/2T
3/2
2 + 6β2
4y
3T2 − β3
4y
6 − δ = 0, (9.3)
Doubly Exotic Nth-Order Superintegrable Classical Systems Separating 13
where δ is an arbitrary constant. In the case δ = 0, we immediately obtain the particular
solutions
T1(x) = β4x
3,
β4
9
x3, and T2(y) = β4y
3,
β4
9
y3,
which correspond to a well-known lower-order superintegrable system.
V1(x)
3
2
1
−1
x
−1.5 −1.0 −0.5 0.5 1.0 1.5
Figure 3. Case N = 5: the x-component V1(x) of the doubly exotic potential V (x, y) = V1(x)+V2(y) of
type Q1. It corresponds to the fifth-order integral Y5 (9.2). From the algebraic equations (9.3) we obtain
the four solutions V1,i(x) = T ′
1,i, i = 1, 2, 3, 4, displayed above. In the case Q1 the y-component V2(y) is
of the same form with four similar solutions V2,i(y). The values β4 = δ = 1 were used.
V = V1(x) + V2(y)
−1.25
−0.1
−0.75
−0.5
−0.25
0
0.25
0
0.2
0.4
x 0.6
0.8
1.0
0
0.2
0.4 y
0.6
0.8
1.0
Figure 4. A doubly exotic potential Q1 corresponding to N = 5. It admits the fifth-order integ-
ral Y5 (9.2). It also possesses bounded trajectories which by construction are closed and periodic. The
values β4 = δ = 1 were used.
The algebra generated by the integrals takes the form
{C,X}PB = −24β4Y(Q1)
5 ,{
C,Y(Q1)
5
}
PB
= 12X
(
H2 −X 2
)2 − 48δXH.
14 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
In the corresponding quantum system analyzed in [1], the case (i) splits into two subclasses
of integrals Y5 that solely differ in their lower order ℏ-dependable terms. Consequently, two
systems called Q1 and Q2 occur. However, in the classical limit ℏ → 0 the two systems Q1
and Q2 coincide.
Next, within case (ii) the classical systems Q3 (A023A050A005 ̸= 0, A122 = A032 = 0) and Q4
(A023A005 ̸= 0, A122 = A050 = A032 = 0) are not superintegrable (like in the quantum case).
Case (ii).
• The systemQ5: A023 ̸= 0, A122 = A032 = A005 = 0, A050 arbitrary. This system corresponds
to A023 = 1 and arbitrary A050, all other Aijk = 0. Again, by solving all the DE (8.2)–(8.4) we
arrive to the first-order nonlinear ODE for T1
5A050(T
′
1)
3 − 12τ2xT ′
1 + 3β1(T
′
1)
2 − 12τ2T1 + µ = 0, (9.4)
τ ̸= 0, β1 and µ are real constants, whereas
V2 ≡ T ′
2 = ±2τ
√
−y.
The corresponding highest-order integral of motion reads
Y(Q5)
5 = A050p
5
1 + p21p
3
2 + p31(5A050T
′
1 + β1) + p1
(
15
2
A050(T
′
1)
2 + 3β1T
′
1 − 6xτ2
)
+ 6τ
√
−yp2p
2
1 + 2T ′
1p
3
2 + 12τ
√
−yT ′
1p2. (9.5)
Clearly, the case A032 = 1 and arbitrary A005 (all other Aijk = 0) also leads to a superintegrable
potential. It can simply be obtained by replacing A050 → A005 and making the permutation
x ⇔ y (V1 ⇔ V2) in (9.4) and (9.5).
V1(x)
4
2
−2
−4
−6
x
−10 −5 5 10
Figure 5. Case N = 5: the x-component V1(x) of the doubly exotic potential V (x, y) = V1(x) + V2(y)
of type Q5. It admits the fifth-order integral Y5 (9.5). From the equation (9.4) we obtain the three
numerical solutions V1,i(x) = T ′
1,i, i = 1, 2, 3, displayed above. The values A050 = 1
5 , τ = 1√
12
, β1 = 1
and µ = −3 were used.
In this case, the algebra generated by the integrals C =
{
Y(Q5)
5 ,X
}
PB
, Y(Q5)
5 and X takes
the form
{C,X}PB = 0,{
C,Y(Q5)
5
}
PB
= −144τ4(H+ X ),
and it does not provide further information about the solutions of (9.4).
Doubly Exotic Nth-Order Superintegrable Classical Systems Separating 15
• The system Q6: A023A032 ̸= 0, A122 = A050 = A005 = 0. This system corresponds to
A023A032 ̸= 0, all other Aijk = 0.
3A032(T
′
1)
2 − σ2xT
′
1 − σ2T1 = 0,
3A023(T
′
2)
2 + σ2yT
′
2 + σ2T2 = 0, (9.6)
σ2 ̸= 0 is a real constant.
Y(Q6)
5 = A032p
2
2p
3
1 +A023p
3
2p
2
1 + 2p32A023T
′
1 + 2p31A032T
′
2 + p1p
2
2
(
3A032T
′
1 −
σ2x
2
)
+ p21p2
(
3A023T
′
2 +
σ2y
2
)
+ p1
(
3xA023T
′
2T
′′
2 + 6A032T
′
1T
′
2 +
1
2
σ2xyT
′′
2
)
+ p2
(
3yA032T
′
1T
′′
1 + 6A023T
′
1T
′
2 −
1
2
σ2xyT
′′
1
)
. (9.7)
V = V1(x) + V2(y)
−0.4
−0.3
−0.2
−0.1
0
0.1
−5 −4
x
−3 −2
−5
−4
−3
−2
y
Figure 6. A doubly exotic potential Q6 corresponding to N = 5. It admits the fifth-order inte-
gral Y5 (9.7). It also possesses bounded trajectories which by construction are closed and periodic. The
values A032 = −A023 = 1
12 , σ1 = 1 were used.
In this case, the algebra generated by the integrals C =
{
Y(Q6)
5 ,X
}
PB
, Y(Q6)
5 and X takes
the form
{C,X}PB = 0,{
C,Y(Q6)
5
}
PB
= 2Xσ2
2
(
H2 −X 2
)2
,
and it does not provide further information about the solutions of (9.6). However, it is easy to
check that
T1(x) =
W 2(x)− σ2x
2
12A032
, T2(y) = −W 2(y)− σ2y
2
12A023
,
satisfy (9.6), where W = W (z) is given by the following third order polynomial equation(
W − z
√
σ2
)(
W + 2z
√
σ2
)2
+ τ = 0,
here τ ̸= 0 is an integration constant.
• The system Q7. This system is a particular case of system Q5. It corresponds to the
situation where A023 = 1 and all other Aijk = 0.
3β1(T
′
1)
2 − 12τ2xT ′
1 − 12τ2T1 + µ = 0, V2 ≡ T ′
2 = ±2τ
√
−y,
τ ̸= 0, β1 and µ are real constants.
Y(Q7)
5 = p21p
3
2 + p31β1 + p1
(
3β1T
′
1 − 6xτ2
)
+ 6τ
√
−yp2p
2
1 + 2T ′
1p
3
2 + 12τ
√
−yT ′
1p2.
16 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
Case (iii).
• The system Q8: A
2
050 +A2
005 ̸= 0, A122 = A023 = A032 = 0. This system corresponds to the
case A2
050 +A2
005 ̸= 0 and all other Aijk = 0.
A050T
′
1
3 + β1T
′
1
2 + θ1T
′
1 − Λx+ κ1 = 0,
A005T
′
2
3 + α1T
′
2
2 + ϕ1T
′
2 + Λy + ω1 = 0, (9.8)
Λ ̸= 0, α1, β1, θ1, κ1, ϕ1 and ω1 are real constants.
Y(Q8)
5 = A050p
5
1 +A005p
5
2 + p31
(
5A050T
′
1 +
5β1
3
)
+ p32
(
5A005T
′
2 +
5α1
3
)
+ p1
(
15
2
A050T
′
1
2 + 5β1T
′
1 +
5θ1
2
)
+ p2
(
15
2
A005T
′
2
2 + 5α1T
′
2 +
5ϕ1
2
)
.
In this case, for the solutions of (9.8) the Poisson bracket C =
{
Y(Q8)
5 ,X
}
PB
∝ λ. Thus, the
algebra of the integrals of motion takes the form
{C,X}PB = 0,
{
C,Y(Q8)
5
}
PB
= 0.
Again, the systemQ8 was found in [28] by means of Heisenberg-type higher order symmetries. All
three conserved quantities
(
H,X ,Y(Q8)
5
)
admit separation of variables in Cartesian coordinates.
The particular case with β1 = θ1 = κ1 = α1 = ϕ1 = ω2 = 0, thus V1 ∝ x
1
3 and V2 ∝ y
1
3 , was
studied in [24] using action-angle variables.
• The system Q9. This system is a particular case of system Q8. It corresponds to the
situation where A050 = 1 and all other Aijk = 0.
A050T
′
1
3 + β1T
′
1
2 + θ1T
′
1 − Λx+ κ1 = 0,
α1T
′
2
2 + ϕ1T
′
2 + Λy + ω1 = 0,
Λ ̸= 0, β1, θ1, κ1, ϕ1, ω1 and α2
1 + ϕ2
1 ̸= 0 are real constants.
Y(Q9)
5 = A050p
5
1 + p31
(
5A050T
′
1 +
5β1
3
)
+
5α1
3
p32 + p1
(
15
2
A050T
′
1
2 + 5β1T
′
1 +
5θ1
2
)
+ p2
(
5α1T
′
2 +
5ϕ1
2
)
.
10 Conclusions
We considered Nth-order superintegrable classical systems in a two-dimensional Euclidean space
separating in Cartesian coordinates. They are characterized by three polynomial (in momentum
variables) integrals of motion (H,X ,YN ). Let us summarize the main results reported in this
paper:
1. Higher-order (N > 2) superintegrable classical systems
H =
1
2
(
p21 + p22
)
+ V1(x) + V2(y),
can be classified into three classes: doubly standard, singly exotic and doubly exotic
potentials. This classification is based on the nature of the equation that defines the most
general form of the potential functions V1(x) and V2(x). For doubly standard potentials
this equation is a linear compatibility condition necessary for the existence of the Nth
order integral of motion YN (in general a PDE of order N in two variables) whilst in the
case of doubly exotic potentials it is given by a nonlinear compatibility condition.
Doubly Exotic Nth-Order Superintegrable Classical Systems Separating 17
2. From the equation {YN ,H}PB = 0, we show in a systematic manner how to find and
successively solve a “well” of NLCC separately for V1(x) and V2(y). It was also indicated
that requiring the integrals of motion (C = {X ,YN}PB,X ,YN ) to span a closed polynomial
algebra may help to simplify (reduce the order) of the DE and eventually to find the explicit
solutions V1(x) and V2(y).
3. All fifth-order (N = 5) superintegrable doubly exotic potentials were derived explicitly
by solving the set of DE. The DE lead to first order non-linear ODEs that define the
functions V1 and V2, respectively. Unlike the quantum case, these equations do not have
the Painlevé property. This was verified either by finding their general solutions explicitly
or by applying a standard test to them [2]. Interestingly, at N = 5 doubly exotic confining
potentials appear for the first time. At N = 4 no doubly exotic potentials occur at all.
4. The present study suggests to explore the inverse problem, namely we take two polyno-
mial functions (A and B) in momentum variables (p1, p2) and construct the new object
C = {A,B}PB. If the algebra generated by (A,B, C) is a closed polynomial algebra with
polynomial coefficients in H, then under what conditions this closure relations imply that A
and B are integrals, i.e., they Poisson commute with H?
Finally, a direct computation for the next two cases N = 6, 7 leads us to the following
conjecture:
Conjecture. There exists an infinite family of N th-order superintegrable systems with an inte-
gral
Y(Doubly exotic)
N = L(N−4)
z p21p
2
2 + (lower order terms), N ≥ 5,{
Y(Doubly exotic)
N ,H
}
PB
= 0.
The associated potential V can be written as follows
V = V1(x) + V2(y) = G′(x;N) + G′(y;N),
here G = G(u;N) obeys a nonlinear first-order ODE of the form
G′[6uN−4G′ + 4(N − 5)uN−5G + F1(u) + σuN−2
]
+ G
[
2(N − 5)uN−6G + F2(u)− 2σuN−3
]
+ F3(u) + buN = 0. (10.1)
The three functions Fq’s in (10.1) are polynomials in the variable u of degree at most (N − 1),
and σ, b are real parameters as well. The equation (10.1) is in complete agreement with the
limit ℏ → 0 of its quantum analogue treated in [13]. In future work, we plan to establish in detail
under what conditions the closed algebra of the integrals of motion is polynomial, and how to use
it as a new systematic tool to solve the determining equations in a simpler and more efficient
manner.
Acknowledgments
İY and AMER during a sabbatical leave and a postdoctoral academic stay at the Centre de
Recherches Mathématiques, Université de Montréal, respectively, were introduced to the subject
of higher-order superintegrability by Pavel Winternitz. His enormous influence is present in this
study as it does in the whole subject. It is with admiration and great affection that we dedicate
this paper to his memory. We thank the anonymous referees and the editor for their valuable
comments and constructive suggestions on the manuscript.
18 İ. Yurduşen, A.M. Escobar-Ruiz and I. Palma y Meza Montoya
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1 Introduction
2 Superintegrability: existence of an Nth-order polynomial integral
2.1 General form
2.2 The determining equations
3 Superintegrable potentials separable in Cartesian coordinates
3.1 The linear compatibility condition
4 The first nonlinear compatibility condition
5 Doubly exotic potentials
5.1 Integral Y_N for doubly exotic potentials
6 ODEs versus algebraic equations. Algebras of integrals of motion
7 Lowest order cases N=3 and N=4: doubly exotic potentials
7.1 Case N=3
7.2 Case N=4
8 Case N=5: doubly exotic potentials
8.1 Determining equations
8.2 The (first) NLCC
9 Results
9.1 Superintegrable potentials
10 Conclusions
References
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| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Yurduşen, İsmet Escobar-Ruiz, Adrián Mauricio Palma y Meza Montoya, Irlanda 2026-01-07T13:41:10Z 2022 Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates. İsmet Yurduşen, Adrián Mauricio Escobar-Ruiz and Irlanda Palma y Meza Montoya. SIGMA 18 (2022), 039, 20 pages 1815-0659 2020 Mathematics Subject Classification: 70H06; 70H33; 70H50 arXiv:2112.01735 https://nasplib.isofts.kiev.ua/handle/123456789/211629 https://doi.org/10.3842/SIGMA.2022.039 Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space 𝐸₂ are explored. The study is restricted to Hamiltonians allowing separation of variables 𝑉(𝑥, 𝑦) = 𝑉₁(𝑥) + 𝑉₂(𝑦) in Cartesian coordinates. In particular, the Hamiltonian ℋ admits a polynomial integral of order 𝑁 > 2. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of the formalism are non-linear compatibility conditions and the algebra of the integrals of motion. The case 𝑁 = 5, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case 𝑁 > 2 and a formulation of the inverse problem in superintegrability are briefly discussed as well. İY and AMER, during a sabbatical leave and a postdoctoral academic stay at the Centre de Recherches Mathématiques, Université de Montréal, respectively, were introduced to the subject of higher-order superintegrability by Pavel Winternitz. His enormous influence is present in this study as it is in the whole subject. It is with admiration and great affection that we dedicate this paper to his memory. We thank the anonymous referees and the editor for their valuable comments and constructive suggestions on the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates Article published earlier |
| spellingShingle | Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates Yurduşen, İsmet Escobar-Ruiz, Adrián Mauricio Palma y Meza Montoya, Irlanda |
| title | Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates |
| title_full | Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates |
| title_fullStr | Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates |
| title_full_unstemmed | Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates |
| title_short | Doubly Exotic 𝑁th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates |
| title_sort | doubly exotic 𝑁th-order superintegrable classical systems separating in cartesian coordinates |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211629 |
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