Acta Polytechnica https://doi.org/10.14311/AP.2025.65.0520 Acta Polytechnica 65(5):520–533, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague CLASSICAL AND QUANTUM SUPERINTEGRABLE SYSTEMS ON THE SPHERE AND THE HYPERBOLIC 2-SPACE Mariano A. del Olmoa,∗, Álvaro Romaniegab a Universidad de Valladolid, Departamento de Física Teórica, Atómica y Optica and IMUVA, Paseo Belén 7, 47011 Valladolid, Spain b Universidad de las Hespérides, C. los Balcones 10, 35001 Las Palmas de Gran Canaria, Spain ∗ corresponding author: marianoantonio.olmo@uva.es Abstract. We present two superintegrable Hamiltonian systems in two dimensions, defined on the sphere and on the hyperbolic plane. These systems are generalised à la Tremblay-Turbiner-Winternitz (TTW), involving the introduction of a real parameter k > 0, with the aim of extending superintegrable Hamiltonian systems to curved spaces in a way similar to the TTW system on the plane. We carry out both classical and quantum analyses of these new systems. We prove that the superintegrability of the initial systems (i.e. when k = 1) is preserved when k is rational, as in the TTW case. A detailed study of their classical counterparts and trajectories is also included. Keywords: Superintegrable systems, factorisation of Hamiltonians, Tremblay-Turbiner-Winternitz Hamiltonan systems. 1. Introduction In [1], a family of superintegrable systems defined on a homogeneous space of the pseudo-orthogonal Lie group O(p, q) was introduced. This work increased the number of known superintegrable systems at the time [2]. This family of superintegrable Hamiltonian systems (SHSs) has since been studied from various perspectives [3–8]. In 2010, Tremblay, Turbiner, and Winternitz intro- duced the well-known TTW integrable system [9, 10], which generalises the Smorodinsky-Winternitz super- integrable system [11, 12]. Its introduction renewed the scientific community’s interest in superintegrable systems, and the number of new SHSs has steadily grown since then [13–18]. The classical TTW system [9, 10] is characterised by the Hamiltonian: H = p2 r + 1 r2 p 2 ϕ +ωr2 + k2 r2 ( a cos2 kϕ + b sin2 kϕ ) , (1) where r ∈ (0,∞), 0 < ϕ < π 2k ; k, ω, a, b are real numbers with k, ω ≠ 0, and a, b > 0, respectively. The quantum version of this system is: H = −∂2 r − 1 r ∂r + ωr2 +k2 r2 ( − 1 k2 ∂ 2 ϕ + a cos2 kϕ + b sin2 kϕ ) . (2) Now, by making the change of variables k ϕ = θ, and replacing a and b with α2 − 1 4 and β2 − 1 4 , respectively, (as in [19]), the Hamiltonian (2) becomes: H = −∂2 r − 1 r ∂r + ωr2 +k2 r2 ( −∂2 θ + α2 − 1 4 cos2 θ + β2 − 1 4 sin2 θ ) , (3) where 0 < θ < θ 2 and α2, β2 > 1 4 . In this paper, we present a generalisation à la TTW of two superintegrable Hamiltonian systems defined in two-dimensional curved spaces, specifically the sphere S2 and the hyperbolic plane H2. A direct extension of the original TTW system to two-dimensional spherical and hyperbolic spaces can be found in [20–23]. The structure of the paper is as follows: in Section 2, we introduce the original system defined on S2, as well as its TTW-type generalisation. In addition, we present a mathematical overview of the factorisation method for Hamiltonians – originating in the work of Schrödinger – which allows us to construct generalised ladder and shift operators. We also present a theorem that establishes a systematic method for constructing two symmetries (or integrals of motion) that com- mute with the Hamiltonian, thereby demonstrating the superintegrability of the new system. Section 3 is devoted to the explicit construction of these sym- metries, which are obtained by factorising two sub- Hamiltonians derived from the original system through the separation of variables method. In Section 4, we study the classical counterpart of this Hamiltonian and derive its classical trajectories in an algebraic way. The hyperbolic case is presented in Section 5, where we follow an analogous approach, as the Hamiltonian systems exhibit formal similarities. We conclude with some final remarks and an appendix, where we present a more general version of Theorem 1 from Section 2. 2. TTW SO(3)-Hamiltonian Let us consider the Hamiltonian [6]: H := − 2∑ i=0 J2 i + l2i − 1 4 s2 i , (4) 520 https://doi.org/10.14311/AP.2025.65.0520 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 65 no. 5/2025 Classical and Quantum Superintegrable Systems on the Sphere . . . where (si) ≡ (s0, s1, s2) ∈ R3 verifies s2 0 + s2 1 + s2 2 = 1 and li ∈ R. The differential operators Ji := εijk sj ∂sk (εijk is the Levi-Civita symbol) span the Lie algebra so(3) since they are the infinitesimal generators in the vector field representation. It is worth noting that ∑2 i=0 J 2 i is the quadratic Casimir operator of so(3), which is the Laplace- Beltrami operator on the sphere S2, which is, in turn, the orbit (symmetric space) of SO(3). Considering spherical coordinates (ϕ1, ϕ2) such that: s0 = cosϕ2 cosϕ1, s1 = cosϕ2 sinϕ1, s2 = sinϕ2, (5) the Hamiltonian (4) becomes: H = − ∂2 ϕ2 + tanϕ2 ∂ϕ2 + l22 − 1 4 sin2 ϕ2 + 1 cos2 ϕ2 [ −∂2 ϕ1 + l20 − 1 4 cos2 ϕ1 + l21 − 1 4 sin2 ϕ1 ] , (6) with 0 < ϕ1 < π 2 , 0 < ϕ2 < π 2 . Now we modify this Hamiltonian following the TTW-Hamiltonians (2) and (3) obtaining a one-parameter family of Hamilto- nians depending on k ∈ R∗: Hk = − ∂2 ϕ2 + tanϕ2 ∂ϕ2 + l22 − 1 4 sin2 ϕ2 + k2 cos2 ϕ2 [ −∂2 kϕ1 + l20 − 1 4 cos2 kϕ1 + l21 − 1 4 sin2 kϕ1 ] , (7) with now 0 < ϕ1 < π 2k , 0 < ϕ2 < π 2 . To simplify the notation, we introduce the change of variables (kϕ1, ϕ2) → (θ, ϕ) such that 0 < θ, ϕ < π 2 [19]: Hk = − ∂2 ϕ + tanϕ∂ϕ + l22 − 1 4 sin2 ϕ + k2 cos2 ϕ [ −∂2 θ + l20 − 1 4 cos2 θ + l21 − 1 4 sin2 θ ] . (8) Note that we actually have a family of Hamiltonians depending on four real parameters (k, l0, l1, l2 ). Taking into account the variable separation, i.e. Ψ(θ, ϕ) = ψ(θ)φ(ϕ), the well-known eigenvalue equation Hk Ψ = EΨ splits in two equations: Hϕ Mk φ(ϕ) = E φ(ϕ), Hθψ(θ) = E′ ψ(θ), (9) where: Hϕ Mk = −∂2 ϕ + tanϕ∂ϕ + l 2 2 − 1 4 sin2 ϕ + M2 k cos2 ϕ = Hϕ + M2 k cos2 ϕ , (10) Hθ = −∂2 θ + l20 − 1 4 cos2 θ + l21 − 1 4 sin2 θ , (11) with Mk := k √ E′ the separation constant. In the following we will use β instead E′, such that β2 := E′, hence Mk := kβ. Thus the new Hamiltonian (8) becomes: Hk = Hϕ Mk + k2(Hθ − β2) cos2 ϕ . (12) 2.1. Hamiltonian factorisation As it is well know from the paper by Infeld and Hull [24] in order to construct the shift operators, we factorise the Hamiltonian [25, 26]. Let {Hm}m∈Z be a family of Hamiltonians such that ∀m ∈ Z: Hm = A+ mA − m + λm = A− m−1A + m−1 + λm−1. (13) Then, it is straightforward to prove that A± m are shift operators (13), verifying: A− m : Hm → Hm+1, A+ m : Hm+1 → Hm, (14) where by Hm, we design the eigenfunction space of the Hamiltonian Hm (as differential operators). In addition, the operators A± m are intertwining op- erators, i.e.: A+ m Hm+1 = Hm A+ m, A− m Hm = Hm+1 A − m. (15) An important consequence of this result is that we can obtain the eigenvectors and the eigenvalues of the discrete spectrum of the original Hamiltonian knowing the ground states of the Hamiltonians Hm of the family. Effectively, let us suppose that ψ0 (m) are the ground states of the Hamiltonians Hm of the hierarchy. They are determined by the condition: A− mψ 0 (m) = 0, (16) hence from Equation (13): Hm ψ0 (m) = ( A+ m A− m + λm ) ψ0 (m) = λmψ 0 (m). (17) Thus, the scalars λm appearing in the factorisation are the energies of the grounds states. Then, defining: m∏ i=0 A+ i := A+ 0 · · ·A+ m, m ∈ N, (18) and taking into account (15), by induction, we obtain: H0 m∏ i=0 A+ i = ( m∏ i=0 A+ i ) Hm+1. (19) Therefore, we conclude that the mth excited eigenfunc- tion ψm (0) of the original Hamiltonian H0 is obtained by the consecutive application of the operators A+ over the ground state ψ0 (m) of Hm: ψm (0) = m−1∏ i=0 A+ i ψ 0 (m). (20) Hence, the eigenvalues of the Hamiltonian H0 are obtained in an algebraic way. 521 Mariano A. del Olmo, Álvaro Romaniega Acta Polytechnica 2.2. Higher rank ladder/shift operators Let us begin by considering a family of Hamiltonians {Hm}m∈I , and introduce some general definitions concerning two types of operators: • Ladder operators, which connect eigenvectors of a fixed Hamiltonian Hk (with k ∈ I fixed), corre- sponding to different eigenvalues. • Shift operators, which connect eigenvectors of dif- ferent Hamiltonians within the family {Hm}m∈I , corresponding to the same eigenvalue. Let Hm be a Hamiltonian, Hm the Hilbert space spanned by its eigenvectors and ψm λ an eigenvector of Hm with eigenvalue λ. An operator L±n (with n ∈ N) satisfying: L±n : Hm −→ Hm ψm λ 7−→ ψm λ±n = L±nψ m λ (21) is called a generalised ladder operator. The standard ladder operator corresponds to the case n = 1. An operator S±n (with n ∈ N) satisfying: S±n : Hm −→ Hm±n ψm λ 7−→ ψm±n λ = S±ψ m λ (22) is called a generalised shift operador. To summarise: Ladder operators connect eigenstates of a single Hamiltonian with different eigenvalues. Shift oper- ators connect eigenstates of different Hamiltonians within the family {Hm} with the same eigenvalue. 2.3. Superintegrability of the TTW SO(3)-Hamiltonian By combining both types of operators in a suitable way, we can construct a symmetry of the Hamiltonian system – namely, a finite differential operator X that commutes with the Hamiltonian [27–29]. At the clas- sical level, these symmetries correspond to integrals of motion, also known as constants of motion. Theorem 1. Let Hk be the Hamiltonian given in Equation (12). Suppose there exist ladder and shift operators as defined in Equations (21) and (22), re- spectively: ψβ ∈ Hθ L±2n−−−→ n∈N∗ ψβ±2n ∈ Hθ, φMk ∈ Hϕ Mk S±2m−−−−→ m∈N∗ φMk±2m ∈ Hϕ Mk±2m, (23) where ψβ ∈ Hθ is an eigenvector of Hθ with eigen- value β2 and φMk ∈ Hϕ Mk is an eigenvector of Hϕ Mk as given in (10), such that k β = Mk. Then, if k = m n , there are two symmetry operators (X±) of the Hamil- tonian Hk, (i.e. operators that commute with Hk), defined as: X± := L±2n S±2m, m, n ∈ N∗ ≡ N − {0}. (24) Since there are 2 × 2 − 1 independent symmetries, namely X± and Hk, the system is maximally super- integrable. Proof. We only need to prove that the two opera- tors X± defined in Equation (24) commute with the Hamiltonian, i.e.: [Hk, X ±] = 0, ∀Mk, β, E, (25) as shown in [28]. Let us consider the eigenfunction φMk ψβ of Hk with eigenvalue E. Then: [Hk, X ±]φMk ψβ = (Hk − E)X±φMk ψβ . (26) On the other hand, from Equations (23) and (24), we have: X±φMk ψβ = ψβ±2nφMk±2m. (27) The Hamiltonian Hk, using Equations (10) and (12), can be rewritten as: Hk = Hϕ + Mk 2 cos2 ϕ + k2(Hθ − β2) cos2 ϕ . (28) Since Mk 2 = (Mk ±2m)2 −(2m)2 ∓4Mk m, we obtain: Hk = Hϕ Mk±2m − (2m)2 cos2 ϕ ∓ 4Mkm cos2 ϕ + k2(Hθ − β2) cos2 ϕ . (29) From Equations (27) and (29), it follows that: Hk X ±φMk ψβ = EφMk±2mψβ±2n, (30) where we used the relation Mk = kβ together with the rationality condition k = m n . By substitut- ing Equations (27) and (30) into (26), we find that [Hk, X ±] = 0, as claimed. 3. Analysis of the TTW SO(3)-Hamiltonian We will use the theory of Hamiltonian factorisa- tion [24] to study the TTW SO(3)-Hamiltonian given in Equation (8). 3.1. Factorisation of Hθ Let us start with the Hamiltonian Hθ (11), which is a trigonometric Pöschl-Teller Hamiltonian [30]: Hθ = −∂2 θ + l20 − 1 4 cos2 θ + l21 − 1 4 sin2 θ . (31) The corresponding operators A± n and λn (13), relative to Hθ are [6]: A± n = ± ∂θ − (l0 + n+ 1 2) tan θ + (l1 + n+ 1 2) cot θ, λn ≡E′ n ≡ β2 = (l0 + l1 + 2n+ 1)2. (32) 522 vol. 65 no. 5/2025 Classical and Quantum Superintegrable Systems on the Sphere . . . The hierarchy of Hamiltonians Hθ n obtained by us- ing Equation (13) have the same expression of Hθ (Equation (11)), that it is the Hamiltonian with n = 0 of the hierarchy, after the changes l0 → l0 + n and l1 → l1 + n. The hierarchy of Hamiltonians Hθ n, defined via Equation (13), all share the same functional form as Hθ in Equation (11) – which corresponds to the case n = 0, up to the parameter shifts l0 → l0 + n and l1 → l1 + n. The fundamental states of Hθ n are obtained by Equa- tion (16) and they are: ψ0 (n)(θ) = cosl0+n+ 1 2 θ sinl1+n+ 1 2 θ, (33) and the excited states of the initial Hamiltonian Hθ ≡ Hθ 0 are given by: ψn (0)(θ) = N cosl0+ 1 2 θ sinl1+ 1 2 θ P (l0,l1) n (cos 2θ), (34) where N is the normalisation constant and P (l0,l1) n are Jacobi polynomials. The energy of these states is E′ n = β2 = λn as given in Equation (32). From [8], the ladder operators are: L± β := ± (β ± 1) sin 2θ ∂θ + β(β ± 1) cos 2θ − l20 + l21. (35) They act as: Hθ ∋ ψβ L+ β−−→ ψβ+2 ∈ Hθ, Hθ ∋ ψβ+2 L− β−−→ ψβ ∈ Hθ. (36) We can define generalised ladder operators as: L+ β→β+2n := 0∏ i=2(n−1) L+ β+i, L− β→β−2n := 2∏ i=2n L+ β−i, (37) that act as: Hθ ∋ ψβ L+ β→β+2n−−−−−−→ ψβ+2n ∈ Hθ, Hθ ∋ ψβ L− β→β−2n−−−−−−→ ψβ−2n ∈ Hθ. (38) We now consider index-free operators L± and (L±)n, defined by removing the subscript β from the operators L± β defined in Equation (36), as follows: L+ψβ := L+ β ψβ , L−ψβ := L− β−2ψβ , ∀β. (39) This notation also extends to the generalised ladder operators L± β→β±2n defined in Equation (37):( L+)n ψβ := L+ β→β+2nψβ ,( L−)n ψβ := L− β→β−2nψβ , ∀β. (40) We state the following interesting result and leave its proof to the reader:[√ Hθ, ( L±)n ] = ±2n ( L±)n. (41) 3.2. Factorisation of Hϕ Mk Our task now is to find intertwining operators M± that factorise the Hamiltonian Hϕ Mk given in Equa- tion (10), that is: Hϕ Mk = M+M− + µ. (42) We have identified four families of ladder operators, denoted M±,i with i = 1, 2, 3, 4, which yield the same factorisation of the Hamiltonian, but satisfy different intertwining relations [29]. These families are: Solution 1: M+,1 = ∂ϕ + (kβ − 1) tanϕ+ −2l2 + 1 2 cotϕ, M−,1 = −∂ϕ + kβ tanϕ+ −2l2 + 1 2 cotϕ, µ1 = ( kβ + l2 − 3 2 )( kβ + l2 − 1 2 ) . (43) By computing M−,1 M+,1 +µ1, we get that it is equal to: −∂2 ϕ + tanϕ∂ϕ + (kβ − 1)2 cos2 ϕ + (l2 − 1)2 − 1 4 sin2 ϕ . (44) Thus, the Hamiltonian given in Equation (44) has the same form as Hϕ Mk in Equation (10), but with the parameters kβ and l2 replaced by kβ − 1 and l2 − 1, respectively. Solution 2: M+,2 = ∂ϕ + (kβ − 1) tanϕ+ 2l2 + 1 2 cotϕ, M−,2 = −∂ϕ + kβ tanϕ+ 2l2 + 1 2 cotϕ, µ2 = ( −kβ + l2 + 1 2 )( −kβ + l2 + 3 2 ) . (45) Here, M−,2M+,2 + µ2 gives: −∂2 ϕ + tanϕ∂ϕ + (kβ − 1)2 cos2 ϕ + (l2 + 1)2 − 1 4 ) sin2 ϕ . (46) In this case, the Hamiltonian (45) has the same form as Hϕ Mk in Equation (10), but with the parameters kβ and l2 replaced by kβ − 1 and l2 + 1, respectively. Solution 3: M+,3 = ∂ϕ − (kβ + 1) tanϕ+ 2l2 + 1 2 cotϕ, M−,3 = −∂ϕ − kβ tanϕ+ 2l2 + 1 2 cotϕ, µ3 = ( kβ + l2 + 1 2 )( kβ + l2 + 3 2 ) . (47) In this case, from M−,3M+,3 + µ3, we get: −∂2 ϕ + tanϕ∂ϕ + (kβ + 1)2 cos2 ϕ + (l2 + 1)2 − 1 4 ) sin2 ϕ , (48) where the parameters kβ and l2 are replaced by kβ+1 and l2 + 1, respectively. 523 Mariano A. del Olmo, Álvaro Romaniega Acta Polytechnica Solution 4: M+,4 = ∂ϕ − (kβ + 1) tanϕ+ −2l2 + 1 2 cotϕ, M−,4 = −∂ϕ − kβ tanϕ+ −2l2 + 1 2 cotϕ, µ4 = ( −kβ + l2 − 3 2 )( −kβ + l2 − 1 2 ) . (49) Now, M−,4M+,4 + µ4 gives: −∂2 ϕ + tanϕ∂ϕ + (kβ + 1)2 cos2 ϕ + (l2 − 1)2 − 1 4 ) sin2 ϕ , (50) where the parameters kβ and l2 are replaced by kβ+1 and l2 − 1, respectively. We have obtained eight operators that modify the parameters β k and l2 by ±1, allowing movement in both directions along the kβ and l2 axes. This enables us to construct a hierarchy of Hamilto- nians, denoted by {Hϕ Mk;n,m}n,m∈Z, associated with the initial HamiltonianHϕ Mk in Equation (10), through the repeated application of the intertwining operators M±,3, as described in Equation (13). The elements of this hierarchy are explicitly given by: Hϕ Mk;n,m = − ∂2 ϕ + tanϕ∂ϕ + (Mk + n)2 cos2(ϕ) + (l2 +m)2 − 1 4 sin2(ϕ) , (51) where we have taken into account that Mk = k β. Note that for n = 0 we recover the initial Hamiltonian Hϕ Mk given in Equation (10). The fundamental states of the Hamiltonians Hkβ ϕ,0,m are obtained using Equation (16), and are given by [6]: φ0 (m)(ϕ) = coskβ ϕ, sinl2+m+ 1 2 ϕ, (52) with β = l0 +l1 +n′ + 1 2 , as in Equation (32), where we fix an arbitrary value n = n′ ∈ N (see Subsection 3.1). The excited states of the initial Hamiltonian Hϕ Mk;0,0, obtained by applying Equation (20), are: φm (0)(ϕ) = N cosβ k ϕ sinl2+ 1 2 ϕ × P (l2+ 1 2 ,β k) m (cos 2ϕ), (53) where N is the normalisation constant and P (l0,l1) m are Jacobi polynomials. The energy associated with these states is: Eβ k,m (0) = ( β + l2 + 2m+ 1 2 )( β + l2 + 2m+ 3 2 ) . (54) 3.3. Factorisation of multi-parametric Hamiltonian Given that the Hamiltonians now depend on multiple indices, we shall establish a generalisation of Subsec- tion 2.1, since the original statement does not apply to this extended setting. Proposition 2. Let {Hm}m∈Z|I| be a family of oper- ators depending on a multi-index m = (mi)i∈I , where |I| denotes the cardinality of the index set I. Assume that for all m ∈ Z|I|: Hm = A+ mA − m + λm = A− f(m)A + f(m) + λf(m), (55) where f : Z|I| → Z|I| acts, component-wise, as f(m) = (fi(mi))i∈I with each fi : Z → Z invert- ible on its domain. Then A± m are shift operators (cf. Equation (14)) satisfying: A− m : Hm → Hf−1(m), A+ m : Hf−1(m) → Hm, (56) where Hm is the the eigenfunction space of the Hamil- tonian Hm. Proof. Effectively, let ψE m ∈ Hm be an eigenvector of Hm with eigenvalue E, i.e. Hm ψE m = E ψE m. From Equation (55): Hf−1 i (m) A − m ψE m = A− m (A+ mA − m + λm)ψE m = A− m Hm ψE m = E (A− m ψE m), (57) since f(f−1(m)) = m by definition of f−1. Thus, A− m maps ψE m to an eigenvector of Hf−1(m) with the same eigenvalue. Similarly, for A+ m: Hm A+ m ψE f−1(m) = A+ m(A− mA + m + λm)ψE f−1(m) = A+ m Hf−1(m) ψ E f−1(m) = E(A+ m ψE f−1(m)). (58) This completes the proof. It is worth noting that A± m preserve the eigenvalues. 3.4. On the factorisation of the multi-indexed Hamiltonian Hϕ Mk In the following section, we analyse the solutions aris- ing from the factorisation of the Hamiltonian Hϕ Mk , in the context of the results of the preceding subsection, with the objective of constructing shift operators that facilitate the determination of the symmetries X±. Let us define the generalised operators M±,i n,m in terms of the operators M± i given in Equations (43), (45), (47), and (49) by performing the replacements kβ → kβ + n and l2 → l2 +m with n,m ∈ Z, that is: M±,i n,m := M± i (kβ → kβ + n, l2 → l2 +m), (59) where i = 1, . . . , 4 . From their definition (59) and from the substitu- tions β k → β k + n and l2 → l2 + m in the factori- sation Equations (43)–(50), we can see that for each i ∈ {1, 2, 3, 4}, the operators M±,i n,m act on the hierar- chy of Hamiltonians {Hϕ Mk;n,m}n,m∈Z as follows: M+,i n,m M−,i n,m + µi n,m = Hϕ Mk;n,m, M−,i n,m M+,i n,m + µi n,m = Hϕ Mk;f−1 1,i (n),f−1 2,i (m), (60) 524 vol. 65 no. 5/2025 Classical and Quantum Superintegrable Systems on the Sphere . . . with f−1 1,i , f −1 2,i : Z → Z invertible index maps such that f−1 1,i (r), f−1 2,i (r) : r → r ± 1 depending on i = 1, 2, 3, 4. In other words, their definitions are determined by the index changes dictated by the intertwining Equa- tions (43)–(50). For instance, f−1 1,4 (n) = n + 1 and f−1 2,4 (m) = m−1. Moreover, the action on the spaces of eigenfunctions Hϕ Mk;n,m of the hierarchy Hamiltonians is as follows: M−,i n,m : Hϕ Mk;n,m → Hϕ Mk;f−1 1,i (n),f−1 2,i (m), M+,i n,m : Hϕ Mk;f−1 1,i (n),f−1 2,i (m) → Hϕ Mk;n,m. (61) We have identified eight operators M±,i n,m that trans- form the eigenstates according to: M±,i n,m : Hϕ Mk;n,m → Hϕ Mk;n+a,m+b, (62) where a, b ∈ {−1,+1}. In Figures 1 and 2, we illustrate the action of the operators M±,i 0,0 ≡ M±,i, which coincide with those defined in Equations (43), (45), (47), and (49). Note that, for our purposes, it is sufficient to consider only one set, either the M+ or the M− operators, as they act in a similar manner. We can define index-free operators Ma,b in terms of, for instance, the operators M−,i m,n (for all n,m ∈ Z) as follows: Ma,b ψm,n := M−,i m,n ψm,n, a, b = ±1. (63) Thus, from Figure 1 and Figure 2 it can be seen that M1,1 = M−,3 m,n, M 1,−1 = M−,2 m,n, M −1,1 = M−,4 m,n, and M−1,−1 = M−,1 m,n. By composing two such opera- tors, we can construct the shift operators S defined in Equation (22), which move only in one direction. In our case, we choose the direction along kβ (i.e. di- rection n). Considering the composition of operators acting as Ma,bψm,n = ψm+a,n+b, we find that: Ma,bM−a,bψm,n = Ma,bψm−a,n+b = ψm,n+2b, (64) where a, b = ±1 . Equation (63) allows us to define the shift operators (22) as follows: S± := Ma,±1 M−a,±1, (65) that act as: S± : Hϕ Mk;m,n → Hϕ Mk;m,n±2. (66) In Figure 3 we show how the shift operator S+ can be expressed in terms of the operators M−1,1 and M1,1. Although S− is not depicted, the reader can infer that it is defined analogously in terms of the operators M−1,−1 and M1,−1. Taking into account Equation (40), we can then ob- tain the shift operators S±2m defined in Equation (23) by: S±2m = ( S±)m . (67) . Figure 1. Action of the operators M±,1 (Equa- tion (43)) and M±,2 (Equation (45)) Figure 2. Action of the operators M±,3 (Equa- tion (47)) and M±,4 (Equation (49)). Figure 3. Shift operator (65) S+ = M1,1 M−1,1. Finally, we have constructed the operators L± ±2n (40) and S±2m (67), which enable us to build the symmetries X± = L±2n S±2m (24) that commute with the Hamiltonian. This construction allows us to prove that the Hamiltonian system Hk (8) is superin- tegrable. It is worth noting that the operators X± do not commute, since [X+, X−] ̸= 0, which is a fact that can be directly verified. Recall that k = m n , with m and n integers, which can, without loss of generality, be taken an irreducible form, i.e. with m and n > 0 coprime. Although k can equivalently be written as k = γm γn for any γ ̸= 0, the operators L±2n and S±2m, depending on θ and ϕ respectively, commute. Using L±2n = (L±)n and S±2m = (S±)m, we obtain L±2nγS±2mγ = (X±)γ , implying that the expression depends only on X±. Thus, it is natural and sufficient to restrict k to its irreducible form. 525 Mariano A. del Olmo, Álvaro Romaniega Acta Polytechnica 4. Associated classical system The classical version of the quantum Hamiltonian (8) is: Hk = p2 ϕ + c2 sin2 ϕ + k2 cos2 ϕ × ( p2 θ + a2 cos2 θ + b2 sin2 θ ) . (68) We can group the terms as: Hθ = p2 θ + a2 cos2 θ + b2 sin2 θ , (69) Hϕ Mk = p2 ϕ + M2 k cos2 ϕ + c2 sin2 ϕ , (70) where Mk := k √ Hθ. Note that Hk = Hk(pϕ, ϕ, pθ, θ) and Hθ = Hθ(pθ, θ). 4.1. Superintegrability In analogy with the quantum case [31], we can consider the ladder functions L±(θ, pθ): L± = ( b2 − a2) 1√ Hθ + cos 2θ √ Hθ ± ipθ sin 2θ, (71) such that they verify: {Hθ, L±}θ = ∓i 4 √ Hθ L±, (72) and also: {Hk, (L±)n} = {Hθ, (L±)n}θ = ∓nα ( L±)n , (73) where: α = 4i √ Hθ k2 cos2 θ = 4i kMk sec2 ϕ. (74) The Poisson brackets {·, ·}θ and {·, ·} refer to the canonical variables (θ, pθ) and (ϕ, pϕ ; θ, pθ), respec- tively. It is worth noting that from Equation (72), we obtain { √ Hθ, L±}θ = ∓i 2 L±, which corresponds to the classical analogue of Equation (41). Similarly, we obtain shift functions S±(ϕ, pϕ) asso- ciated with Solutions 3 and 4 from Subsection 3.1: S± = −l22 cot2 ϕ− (pϕ ∓ iMk tanϕ)2. (75) They verify: {Hϕ Mk ,S±} = ± 4iMk sec2 ϕS±, {H, (S±)m} = ± mα ( S±)m. (76) Now, considering the functions: X± = (S±)m(L±)n, (77) we obtain the following Poisson commutation rela- tions: {H,X±} = 0 if k = m n , (78) which show that X± are integrals of motion. This es- tablishes the superintegrability of the classical system described by Equation (68). 4.2. Classical trajectories From the constants of motion X± (Equation (77)), by substituting Hθ with its expression in terms on Mk = k √ Hθ, we get: X± = ( −l22 cot2 ϕ− (pϕ ∓ iMk tanϕ)2)m × ( b2 − a2√ a2 sec2 θ + b2 csc2 θ + p2 θ + cos 2θ × √ a2 sec2 θ + b2 csc2 θ + p2 θ ± i pθ sin 2θ )n . (79) In this system, H,Hθ,X±, and Mk are constants of motion, but only three are functionally independent. By fixing the total energy H = E, both E and Mk remain constant along the classical trajectory. This allows us to express the generalised momenta in terms of the generalised coordinates: pθ = εθ √ M2 k k2 − ( a2 cos2 θ + b2 sin2 θ ) , pϕ = εϕ √ E − ( l22 sin2 ϕ + M2 k cos2 ϕ ) , (80) where εθ, εϕ ∈ {±1}. The symmetry functions X± are complex-valued, and therefore the constants of motion are, in general, complex numbers C. To obtain physically meaning- ful (real) representations, we make use of the reality condition (X+)∗ = X−, and define: X+ = C, X− = C∗. (81) We then consider the real and imaginary parts of X+: Re(X+) = X+ + X− 2 = Re(C), Im(X+) = X+ − X− 2i = Im(C). (82) These two real functions can be used to describe the trajectories of the system. The classical trajectories T0 are implicitly defined as the set of points x ∈ R3 satisfying the condition: X(x) = C0, (83) where C0 is the fixed complex constant determined by the initial conditions. Figures 4 and 5 show trajectories corresponding to different values of k = m n , as presented [29]. These plots were generated using Mathematica. 5. Hyperbolic TTW SO(2, 1)-Hamiltonian In this case, we consider the Hamiltonian introduced in [7]: H := J2 2 −J2 1 −J2 0 − l 2 2 − 1 4 s2 2 + l21 − 1 4 s2 1 + l20 − 1 4 s2 0 , (84) 526 vol. 65 no. 5/2025 Classical and Quantum Superintegrable Systems on the Sphere . . . (a). m = 1, n = 1. (b). m = 2, n = 1. Figure 4. Trajectories for m = 1, n = 1 and m = 2, n = 1, respectively. (a). m = 1, n = 3. (b). m = 3, n = 3. Figure 5. Trajectories for m = 1, n = 3 and m = 3, n = 3, respectively. where the coordinates of the ambient space (si) ≡ (s0, s1, s2) ∈ R3 satisfy the constraint s2 0+s1 1−s2 2 = −1. The differential operators Ji are given by: J0 = s1∂2 + s2∂1, J1 = s2∂0 + s0∂2, J2 = s0∂1 − s1∂0, (85) and they generate the Lie algebra so(2, 1), with com- mutation relations: [J0, J1] = −J2, [J2, J0] = J1, [J1, J2] = J0. (86) Next, we introduce coordinates analogous to the spher- ical coordinates, denoted by (ξ, θ), to proceed with the analysis: s0 = cos θ sinh ξ, s1 = sin θ sinh ξ, s2 = cosh ξ, (87) where 0 ≤ θ < 2π and 0 ≤ ξ < ∞ . The Hamilto- nian (84) can be rewritten in terms of the variables (ξ, θ) as: H = − ∂2 ξ − coth ξ ∂ξ − l 2 2 − 1 4 cosh2 ξ + 1 sinh2 ξ [ −∂θ + l21 − 1 4 sin2 θ + l20 − 1 4 cos2 θ ] . (88) By deforming this Hamiltonian à la TTW, using the real parameter k ̸= 0 as in Section 2, we arrive at the TTW-Hamiltonian: Hk = − ∂2 ξ − coth ξ∂ξ − l 2 2 − 1 4 cosh2 ξ + k2 sinh2 ξ [ −∂θ + l21 − 1 4 sin2 θ + l20 − 1 4 cos2 θ ] , (89) where 0 ≤ θ < π 2 and 0 ≤ ξ < ∞. In this way we have constructed a family of Hamiltonians {Hk}, depending on four real parameters (k, l0, l1, l2 ). The Hamiltonian may be separated in two “sub-Hamiltonians” through variable separation in the Schrödinger equation Hk Ψ(θ, ξ) = EΨ(θ, ξ) by assuming a factorised solution of the form Ψ(θ, ξ) = ψ(θ)φ(ξ). This leads to two eigenvalue equations: Hθψ = E′ψ, Hξ Mk φ = Eφ, (90) with E′ = β2, and Mk = kβ is the separation constant, where: Hθ := −∂θ + l21 − 1 4 sin2 θ + l20 − 1 4 cos2 θ , (91) Hξ Mk := −∂2 ξ − coth ξ∂ξ − l 2 2 − 1 4 cosh2 ξ + M2 k sinh2 ξ . (92) It is worth noting that the Hamiltonian Hθ (Equation (91)) coincides with the Hamiltonian (Equation (11)) that appeared in the TTW SO(3)- Hamiltonian discussed in Section 2, whose factorisa- tion carried out in Subsection 3.1. 5.1. Factorisation of Hξ Mk We further identify four distinct families of ladder operators N±,i, (i = 1, 2, 3, 4) , analogous to those arising from the TTW SO(3)-Hamiltonian case (Sub- section 3.2): Hξ Mk = N+,iN−,i + µi. (93) As in the spherical case, these operators yield the same factorisation of the Hamiltonian (92), although they differ in their intertwining relations. They are: Solution 1: N+,1 = ∂ξ + ( 1 2 − l2 ) tanh ξ + (1 −Mk) coth ξ, N−,1 = −∂ξ + ( 1 2 − l2 ) tanh ξ −Mk coth ξ, µ1 = −1 4(2l2 + 2Mk − 3)(2l2 + 2Mk − 1). (94) For this solution, we obtain the following expression for N−,1N+,1 + µ1: −∂2 ξ − coth ξ ∂ξ − (l2 − 1)2 − 1 4 cosh2 ξ + (Mk − 1)2 sinh2 ξ . (95) The Hamiltonian given in Equation (95) has the same form as Hξ Mk in Equation (92), but with the parame- ters Mk and l2 replaced by Mk − 1 and l2 − 1, respec- tively. 527 Mariano A. del Olmo, Álvaro Romaniega Acta Polytechnica Solution 2: N+,2 = ∂ξ + ( 1 2 + l2 ) tanh ξ + (1 −Mk) coth ξ, N−,2 = −∂ξ + ( 1 2 + l2 ) tanh ξ −Mk coth ξ, µ2 = −1 4(2l2 − 2Mk + 1)(2l2 − 2Mk + 3). (96) In this case N−,2N+,2 + µ2 yields: −∂2 ξ − coth ξ ∂ξ − (l2 + 1)2 − 1 4 cosh2 ξ + (Mk − 1)2 sinh2 ξ , (97) where the parameters Mk and l2 are replaced by Mk−1 and l2 + 1, respectively. Solution 3: N+,3 = ∂ξ + ( 1 2 + l2 ) tanh ξ + (Mk + 1) coth ξ, N−,3 = −∂ξ + ( 1 2 + l2 ) tanh ξ +Mk coth ξ, µ3 = −1 4(1 + 2l2 + 2Mk)(3 + 2l2 + 2Mk). (98) Here, the expression N−,3N+,3 + µ3 results in: −∂2 ξ − coth ξ ∂ξ − (l2 + 1)2 − 1 4 cosh2 ξ + (Mk + 1)2 sinh2 ξ , (99) where the parameters Mk and l2 are replaced by Mk+1 and l2 + 1, respectively. Solution 4: N+,4 = ∂ξ + ( 1 2 − l2 ) tanh ξ + (Mk + 1) coth ξ, N−,4 = −∂ξ + ( 1 2 − l2 ) tanh ξ +Mk coth ξ, µ4 = −1 4(2l2 − 2Mk − 3)(2l2 − 2Mk − 1). (100) Evaluating N−,4N+,4 + µ4 results in: −∂2 ξ − coth ξ ∂ξ − (l2 − 1)2 − 1 4 cosh2 ξ + (Mk + 1)2 sinh2 ξ , (101) where the parameters Mk and l2 are replaced by Mk+1 and l2 − 1, respectively. We have identified eight operators that shift the parameters β k and l2 by ±1, allowing movement in both directions along the kβ and l2 axes. We also construct a hierarchy of Hamiltonians, {Hξ Mk;n,m}n,m∈Z, associated with the initial Hamilto- nian Hξ Mk (Equation (92)), by the repeatedly applying the intertwining operators N±,3 as described in Equa- tion (13). The elements of this hierarchy are explicitly given by: Hξ Mk;n,m = − ∂2 ξ − coth ξ ∂ξ − (l2 +m)2 − 1 4 cosh2 ϕ + (kβ + n)2 sinh2 ϕ , (102) where we have used the relation Mk = k β. For m = n = 0, we obtain the initial Hamiltonian Hξ Mk given by Equation (92). The fundamental states of the Hamiltonians Hξ Mk;0,m are obtained via Equation (16) and are given by [7]: φ0 (m)(ξ) = coshl2+m+ 1 2 ξ sinhkβ ξ, (103) with β = l0 + l1 + n′ + 1 2 (Equation (32)), where we have taken a fixed, but arbitrary value of n = n′ ∈ N (see Subsection 3.1), The excited states of the original Hamiltonian Hkβ ϕ,0,0, obtained using Equation (32), are [32]: φm (0)(ξ) = N coshl2+ 1 2 ξ sinhβ k ξ × P (l2+ 1 2 ,β k) m (cosh 2ξ), (104) where N is the normalisation constant and P (l0,l1) m are Jacobi polynomials. The energy of these states is: Eβ k,m (0) = − ( βk + l2 + 2m+ 1 2 ) × ( βk + l2 + 2m+ 3 2 ) . (105) We can define generalised operators N±,i n,m in terms of the operators N± i given in Equations (94), (96), (98), and (100), by replacing kβ with kβ + n and l2 by l2 +m with n,m ∈ Z. That is: N±,i n,m := N± i (kβ → kβ + n, l2 → l2 +m), (106) for i = 1, . . . , 4 . Similarly to the sphere case (Subsection 3.4), we can consider index-free operators Na,b defined, for ex- ample, in terms of the operators N−,i m,n for all n,m ∈ Z as follows: Na,b φm,n := N−,i m,n φm,n, a, b = ±1, (107) such that N1,1 = N−,3 m,n, N 1,−1 = N−,2 m,n, N−1,1 = N−,4 m,n, and N−1,−1 = N−,1 m,n. By composing two of these operators, we can construct the shift operators, defined in Equation (22), which move only in a single direction. In our case, we choose the direction along kβ (i.e. the n direction). Considering the composition of these operators act- ing as Na,bφm,n = φm+a,n+b, where φm,n ∈ Hξ Mk;n,m, we obtain that: Na,bN−a,bφm,n = Na,bφm−a,n+b = φm,n+2b, (108) with a, b = ±1 . Equation (107) allows us to define the shift operators (22) as: S± := Na,±1 N−a,±1, (109) such that: S± : Hξ Mk;m,n → Hξ Mk;m,n±2. (110) 528 vol. 65 no. 5/2025 Classical and Quantum Superintegrable Systems on the Sphere . . . Taking into account Equation (40), we can obtain the shift operators S±2m defined in Equation (23) as: S±2m = ( S±)m . (111) Thus, together with the operators L± ±2n (Equa- tion (23)) and S±2m (Equation (111)), we can con- struct the symmetries X± (Equation (15)), that com- mute with the Hamiltonian. This allows us to prove that the Hamiltonian system Hk (Equation (89)) is superintegrable whenever k = m n is a rational number. As mentioned in the TTW SO(3)-Hamiltonian case, we also find that [X+, X−] ̸= 0. By evaluating the double commutators [X±, [X+, X−]], one paves the way for constructing the algebra of integrals of mo- tion. Incidentally, [33] shows that, for both initial Hamiltonians (SO(3) and SO(2, 1), when k = 1, the algebra of integrals of motion coincides with the Racah algebra R(3). Furthermore, in [34] a new algebraic method to describe the symmetry of a quadratically superintegrable system on the two-spherecommonly associated with R(3). Instead of relying on explicit operator realisations, the symmetry algebra is built directly from the enveloping algebra of su(3), using polynomials of degrees 2–4 in a maximal Abelian sub- algebra. This leads to a new six-dimensional cubic algebra with integer structure constants, which in specific realisations reduces to R(3). Moreover, a con- traction of this cubic algebra to the symmetry algebra of a Smorodinsky-Winternitz model on the sphere is shown. In a recent work [35], it was demonstrated that for integer values of the TTW parameter k, the TTW- Hamiltonian, together with two independent integrals of motion and their commutator generate a finite- dimensional polynomial algebra of k + 1 order. This algebra exhibits polynomial, rather than linear, clo- sure and is referred to as the hidden algebra g(k). For k = 1, 2, 3, 4 the polynomial structure has been explicitly established, and it is conjectured that the same holds for all positive integer k. The polynomial degree increases with k, reflecting the higher-order nature of the additional integral of motion. The specific cases we have analysed in this paper fall within this framework and will be the subject of a forthcoming publication elsewhere. 5.2. Associated classical Hamiltonian The classical counterpart of the Hamiltonian (89), including the “coupling constant” k, is: Hk = p2 ξ − l22 cosh2 ξ + k2 Hθ sinh2ξ , (112) where Hθ is given by Equation (69), and Hξ Mk is defined by: Hξ Mk = p2 ξ − l22 cosh2 ξ + M2 k sinh2ξ , (113) with M 2 k = k2 Hθ. It is worth noting that, unlike in the quantum case, it now depends on θ. For Hθ (Equation (69)), we obtained the ladder functions L± (Equation (71)) in Subsection 4.1, and for HMk ξ (Equation (113)) we have the shift functions: S± = (−l2 tanh ξ ∓ Mk coth ξ + ipξ) × (l2 tanh ξ ∓ Mk coth ξ + ipξ). (114) Both operators satisfy: {Hk,S±} = ±α S±, {H, L±} = ∓kα L±, (115) with: α = 4Mk i sinh2 ξ . (116) Then, the functions X± := (S±)m(L±)n satisfy: {Hk,X±} = 0 if k = m n ∈ Q. (117) Thus, for each rational value of k, there exist two independent integrals of motion, explicitly given by: X± = ( M2 k coth2 ξ − l22 tanh2 ξ ∓2iMkpξ coth ξ − p2 ξ )m ×  b2 − a2√ a2 cos2 θ + b2 sin2 θ + p2 θ + cos 2θ × √ a2 cos2 θ + b2 sin2 θ + p2 θ ± ipθ sin 2θ  n , (118) where a, b ∈ R (see Equation (68)), which satisfy the reality condition (X+)∗ = X−. This condition will be used to compute the classical trajectories, similarly to the spherical case. For our system, Hk,Hθ,X±, and Mk are all constants of motion, but only three are functionally independent. By fixing the value of Hk = E, both E and Mk remain constant along the trajectory. Thus, the generalised momenta can be expressed in terms of the generalised coordinates as follows: pθ = εθ √ M2 k k2 − ( α2 cos2 θ + β2 sin2 θ ) , pξ = εξ √ E + ( l22 cosh2 ξ − M2 k sinh2 ξ ) , (119) with εθ, εξ ∈ {±1}. Since the symmetry functions X± are complex, the constants of motion will be complex numbers C. To obtain real representations, and considering the reality condition for X±, we set X+ = C, and X− = C∗, we can consider: Re(X+) = X+ + X− 2 = Re(C), Im(X+) = X+ − X− 2i = Im(C). (120) 529 Mariano A. del Olmo, Álvaro Romaniega Acta Polytechnica (a). m = 1, n = 1. (b). m = 2, n = 1. Figure 6. Trajectories for m = 1, n = 1 and m = 2, n = 1, respectively. Thus, the trajectories T0 are implicitly defined by the set of points x ∈ R3 satisfying: X(x) = C0, (121) where C0 is a fixed complex constant. In Figures 6–8 we present trajectories plotted using Mathematica for various values of k = m n . 6. Conclusions We have examined in detail two new families of Hamil- tonians defined on curved spaces, derived from well- known superintegrable Hamiltonians on the sphere [6] and hyperbolic 2-space [7] through a TTW procedure analogous to that used in [9] to generate the TTW Hamiltonians from the Smorodinsky-Winternitz sys- tem [12]. This procedure involves deforming the initial Hamiltonian by a real parameter k ̸= 0, which recovers the initial system in the limit k → 1. To prove the superintegrability of these new TTW Hamiltonian families, we construct generalised lad- der and shift operators via the factorisation method. These operators yield two symmetries of the TTW Hamiltonian, demonstrating that superintegrability is preserved when k is rational. Furthermore, we study the classical counterparts of these Hamiltonians by following a procedure parallel to the quantum case and guided by the correspondence principle [31]. We derive classical analogues of the ladder and shift quantum operators as classical func- tions, and identify two functions in involution with the Hamiltonian, proving classical superintegrability for rational k. Additionally, the classical trajectories are obtained through an algebraic approach. An open and mathematically significant problem, particularly in light of [33–35], is the explicit deter- mination of the polynomial algebra underlying the integrals of motion in both cases. A deeper under- standing of this algebraic structure could provide new insights into the symmetry properties and superinte- grability of these two Hamiltonian systems. (a). m = 1, n = 2. (b). m = 3, n = 3. Figure 7. Trajectories for m = 1, n = 2 and m = 3, n = 3, respectively. (a). m = 4, n = 4. (b). m = 4, n = 5. Figure 8. Trajectories for m = 4, n = 4 and m = 4, n = 5, respectively. 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Suppose that there exist ladder and shift operators Ly n and Sx m, respectively, acting as Equation (23): Ly ±n : Hy → Hy, Sx ±m : Hx Mk → Hx Mk±m, ψβ(y) 7→ ψβ±n = L±nψβ(y), φMk (x) 7→ φMk±m(x) = Sx ±mφ Mk (x), (123) where ψβ ∈ Hθ is an eigenvector of Hy with eigenvalue β2 and φMk ∈ Hx Mk is an eigenvector of Hx Mk . Then, if k = m n is a rational number, there are two symmetry operators (X±) of the Hamiltonian Hk, (i.e. operators that commute with Hk), defined as: X± := Ly ±n S x ±m, m, n ∈ N∗ ≡ N − {0}. (124) Since there are 2 × 2 − 1 independent symmetries, (namely X± and Hk), the system is maximally superintegrable. Proof. By computing the action of the commutator [Hk, L y nS x m] on an eigenfunction Ψ(x, y) = φMk (x)ψβ(y) of Hk with eigenvalue E, we obtain: [Hk, L y nS x m]φMk ψβ = (Hk − E)Ly nS x mφMk ψβ , (125) since φMk ψβ is an eigenfunction of Hk with eigenvalue E. This follows directly from the nested structure of the Hamiltonian (122) and the fact that Hx Mk φMk = EφMk . By computing Ly nS x mφMk ψβ , we obtain: Ly nS x mφMk ψβ = ψβ+nφMk+m, (126) from the definitions of the ladder and shift operators Ly n (Equation (21)) and Sx m (Equation (13)), where m and n are integers. Taking this fact into account, the Hamiltonian Hk (Equation (122)) takes the form: Hk = Hx + (Mk +m)2 f(x) + −m2 − 2Mkm f(x) + k2(Hy − E′) f(x) = Hx Mk+m + −m2 − 2Mkm f(x) + k2(Hy − E′) f(x) , (127) since Hx + (Mk +m)2/f(x) = Hx Mk+m. Applying the Hamiltonian Hk (Equation (127)) to the function ψβ+nφMk+m (Equation (126)) we find that: HkφMk+mψβ+n = ( E + −m2 − 2Mkm f(x) + k2(E′ + 2βn+ n2 − E′) f(x) ) φMk+mψβ+n. (128) For the commutator in Equation (125) to vanish, it is sufficient that the additional term on the right-hand side of Equation (128) vanishes, namely: −m2 − 2Mkm+ k2(2βn+ n2), (129) which can be rearranged as: −(Mk +m)2 + (Mk + kn)2 = 0, (130) recalling that kβ = Mk. Equation (130) admits two possible solutions: Mk +m = Mk + kn, (131) Mk +m = −Mk − kn. (132) The second solution (Equation (132)) is valid only for specific values of ψβ and Mk, and not in general. Therefore, the appropriate and general solution is the first one, Equation (131), which leads to the condition: k = m n . (133) This result shows that we must construct ladder operators Ly n and shift operators Sx m such that they preserve the quantity Mk. In our two systems, as previously observed, both m and n must be even integers. Moreover, since both operators depends of different variables, they commute. 533 Acta Polytechnica 65(5):520–533, 2025 1 Introduction 2 TTW SO(3)-Hamiltonian 2.1 Hamiltonian factorisation 2.2 Higher rank ladder/shift operators 2.3 Superintegrability of the TTW SO(3)-Hamiltonian 3 Analysis of the TTW SO(3)-Hamiltonian 3.1 Factorisation of H 3.2 Factorisation of HMk 3.3 Factorisation of multi-parametric Hamiltonian 3.4 On the factorisation of the multi-indexed Hamiltonian HMk 4 Associated classical system 4.1 Superintegrability 4.2 Classical trajectories 5 Hyperbolic TTW SO(2,1)-Hamiltonian 5.1 Factorisation of HMk 5.2 Associated classical Hamiltonian 6 Conclusions Acknowledgements References A Generalisation of Theorem 1