Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp Maximally superintegrable systems in flat three-dimensional space are linearizable Cite as: J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 Submitted: 11 March 2020 • Accepted: 27 December 2020 • Published Online: 25 January 2021 M. C. Nucci1,a) and R. Campoamor-Stursberg2,b) AFFILIATIONS 1 Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, & INFN Sezione di Perugia, 06123 Perugia, Italy 2Instituto de Matemática Interdisciplinar and Fac. CC. Matemáticas UCM, Pza. Ciencias 3, E-28040 Madrid, Spain a)Author to whom correspondence should be addressed: nucci@unipg.it b)rutwig@ucm.es ABSTRACT All maximally superintegrable Hamiltonian systems in three-dimensional flat space derived in the work of Evans [Phys. Rev. A 41, 5666–5676 (1990)] are shown to possess hidden symmetries leading to their linearization, likewise the maximally superintegrable Hamil- tonian systems in two-dimensional flat space as shown in the work of Gubbiotti and Nucci [J. Math. Phys. 58, 012902 (2017)]. We conjecture that even minimally superintegrable systems in three-dimensional flat space have hidden symmetries that make them linearizable. Published under license by AIP Publishing. https://doi.org/10.1063/5.0007377 I. INTRODUCTION The superintegrability of (classical) Hamiltonian systems has generally been analyzed from the perspective of separating coordinates of the associated Hamilton–Jacobi equation, an approach that motivated the systematic classification of (orthogonal) coordinate systems in R3 for which a separation of variables can be obtained. For the Hamilton–Jacobi equation in the potential-free case, 11 coordinate systems with the separability property were found, and for each of these classes, Eisenhart determined the most general form of the potential that can be added such that the separability is preserved.1 Although some of these systems, such as the Kepler problem, were recognized to have relevant symmetry properties related to Lie’s approach to differential equations,2,3 the Lie group analysis has not been exploited systematically in the context of superintegrability, in particular, its relation to the linearization problem. Several important classes of superintegrable Hamiltonian systems in flat two-dimensional spaces have been shown in Ref. 4 to be lin- earizable by means of some hidden symmetries of the system using a powerful method originally developed in Ref. 2 in the context of the Kepler problem. This reduction technique, valid for any first-order autonomous system, is based on the observation that one of the unknown functions can always be taken to be the new independent variable. Rewriting the system in these new coordinates and apply- ing the Lie group analysis allow us to determine symmetries that cannot be detected in the original coordinates and therefore correspond to hidden symmetries of the system (see, e.g., Refs. 2 and 4 for details). This procedure further allowed us to show that the hidden lin- earity is completely independent of the degree of the first integrals of the system, as well as the separability properties of the associated Hamilton–Jacobi equation. In Ref. 5, the linearizability of a Hamiltonian system devoid of first integrals quadratic in the momenta but possessing constants of the motion of third- and fourth-order was shown, providing a first example exhibiting hidden linearity but with no second-order integrals. The symmetry analysis in the case of the non-Euclidean plane, considered in Ref. 6, points out that the sym- metry approach remains valid, regardless of the space curvature. The success of the Lie method in linearizing these systems suggests us to inspect higher-order systems along the same lines in an attempt to vindicate the usefulness of the Lie group analysis in the context of superintegrability. The main purpose of this work is to show that the maximally superintegrable Hamiltonian systems in R3 classified in Ref. 7 and possessing linear or quadratic first integrals in the momenta can also be linearized by means of their hidden symmetries. The J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-1 Published under license by AIP Publishing https://scitation.org/journal/jmp https://doi.org/10.1063/5.0007377 https://www.scitation.org/action/showCitFormats?type=show&doi=10.1063/5.0007377 https://crossmark.crossref.org/dialog/?doi=10.1063/5.0007377&domain=pdf&date_stamp=2021-January-25 https://doi.org/10.1063/5.0007377 http://orcid.org/0000-0003-1453-0988 http://orcid.org/0000-0003-2907-8533 mailto:nucci@unipg.it mailto:rutwig@ucm.es https://doi.org/10.1063/5.0007377 Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp main tool used in this context is the Lie criterion that establishes the equivalence between the existence of a point transformation that reduces an nth-order differential equation (or system) to the free equation w(n) = 0 and a Lie algebra of point symmetries of the maximal possible dimension (see Refs. 8 and 9 and references therein). As can be expected, the linearizing transformations are gen- erally nonlinear in their arguments, thus corresponding to a more general class than the canonical transformations that preserve the Hamiltonian. As will be seen, some of the superintegrable Hamiltonian systems in Ref. 7 can be directly linearized, without requiring a change of the independent variable. This approach is sometimes the simplest and computationally shortest among the various possible ways to determine the hidden symmetries. For comparison purposes with the linearization using the reduction, those systems that are solved without changing the independent variable are reconsidered in the Appendix, illustrating that the amount of calculations increases considerably when new independent variables are introduced. On the other hand, for certain systems, it is more convenient to express the Hamiltonian systems in coordinates different from the Cartesian ones, with the purpose of simplifying the symmetry analysis. Indeed, the procedure to linearize a system is far from being unique, and for each of the superintegrable systems, the most efficient way in terms of calculations has been chosen. II. MAXIMALLY SUPERINTEGRABLE SYSTEMS IN R3 In Ref. 7, it was shown that there are five equivalence classes of maximally superintegrable systems in R3 admitting integrals that are at most quadratic in the canonical momenta. If H = 1 2(p 2 1 + p2 2 + p2 3) + V(w1,w2,w3) denotes the Hamiltonian of the system in Cartesian coordinates, then these potentials were given by 1. VI(w1,w2,w3) = k(w2 1 +w 2 2 +w 2 3) + k1 w2 1 + k2 w2 2 + k3 w2 3 , 2. VII(w1,w2,w3) = − k √ w2 1+w 2 2+w 2 3 + k1 w2 1 + k2 w2 2 , 3. VIII(w1,w2,w3) = k1w1 w2 2 √ w2 1+w 2 2 + k2 w2 2 + k3 w2 3 , 4. VIV(w1,w2,w3) = k1w1 w2 2 √ w2 1+w 2 2 + k2 w2 2 + k3w3, 5. VV(w1,w2,w3) = k(w2 1 +w 2 2) + 4kw2 3 + k1 w2 1 + k2 w2 2 , where k, k1, k2, and k3 are arbitrary constants. These Hamiltonian systems with the above potentials are all autonomous so that the general reduction method of Ref. 2 to detect hidden symmetries is potentially applicable to them. We analyze each of these potentials separately and prove that the corresponding Hamiltonian system is linearizable. A. The potential V I (w 1, w 2, w 3) The Hamiltonian H1 = 1 2 (p2 1 + p2 2 + p2 3) + k(w2 1 +w 2 2 +w 2 3) + k1 w2 1 + k2 w2 2 + k3 w2 3 (1) yields the Hamiltonian equations ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ẇ1 = p1, ẇ2 = p2, ẇ3 = p3, ṗ1 = 2 k1 − kw4 1 w3 1 , ṗ2 = 2 k2 − kw4 2 w3 2 , ṗ3 = 2 k3 − kw4 3 w3 3 . (2) We apply the reduction method developed in Ref. 2 to this system. If we choose w1 as a new independent variable y, then system (2) reduces to the following five equations: J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-2 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ dw2 dy = p2 p1 , dw3 dy = p3 p1 , dp1 dy = 2 k1 − ky4 y3p1 , dp2 dy = 2 k2 − kw4 2 w3 2p1 , dp3 dy = 2 k3 − kw4 3 w3 3p1 . (3) The third equation of system (3) can be integrated to yield 2ky4 + 2k1 + p2 1y2 4y2 = A = const. Ô⇒ p1 = ± √ 2 y √ 2Ay2 − ky4 − k1. (4) This coincides with the first integral I1 determined in Ref. 7. Deriving p2 from the first equation of system (3) and replacing it into the fourth equation yield the following second-order equation in w2: d2w2 dy2 = (ky4 − k1)w 3 2 dw2 dy + (k2 − kw4 2)y 3 yw3 2(2Ay2 − ky4 − k1) . (5) Analogously, deriving p3 from the second equation of system (3) and replacing it into the fifth equation give the following second-order equation in w3: d2w3 dy2 = (ky4 − k1)w 3 3 dw3 dy + (k3 − kw4 3)y 3 yw3 3(2Ay2 − ky4 − k1) , (6) which is equal to Eq. (5) if k3 is substituted with k2. Equation (5) admits a three-dimensional Lie symmetry algebra isomorphic to sl(2,R). This suggests us to apply the general method described in Ref. 10, valid for any second-order ordinary differential equation exhibiting sl(2,R) symmetry. If we solve Eq. (5) with respect to the constant k2 and derive it once with respect to y, then we obtain the following third-order equation, d3w2 dy3 = − 3 w2 dw2 dy d2w2 dy2 + 3(ky4 − k1) y2w2(2Ay2 − ky4 − k1) ⎛ ⎝ yw2 d2w2 dy2 + y( dw2 dy ) 2 −w2 dw2 dy ⎞ ⎠ , (7) which admits a seven-dimensional Lie symmetry algebra and is therefore linearizable (see, e.g., Ref. 9, p. 244). A two-dimensional Abelian intransitive subalgebra is that generated by the operators 1 w2 ∂w2 , A − ky2 w2 ∂w2 . (8) Following Lie’s classification (Ref. 8, p. 405), if we transform them into their canonical form, i.e., ∂u, v∂u, then we obtain that the new dependent and independent variables are given by u = w2 2/2, v = A − ky2, and Eq. (7) becomes linear, i.e., d3u dv3 = 3v A2 − kk1 − v2 d2u dv2 . (9) Integration of the latter yields the general solution u = b1 √ A2 − kk1 − v2 + b2v + b3 (10) with bj(j = 1, 2, 3) arbitrary constants. Thus, the general solution of Eq. (7) is J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-3 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp w2 = ± √ 2( √ 2Aky2 − k2y4 − kk1b1 + Ab2 − b2ky2 − b3), (11) which substituted into (5) obviously yields that one of the three arbitrary constants bj is not at all arbitrary and, in particular, depends on k2, i.e., b3 = 1 2 √ k √ 4A2b2 1k + 4 A2b2 2k − 4b2 1k2k1 − 4b2 2k2k1 + k2. (12) Consequently, the general solution of Eq. (5) is w2 = ± ¿ Á ÁÀ2( √ 2Aky2 − k2y4 − kk1b1 + Ab2 − b2ky2 − 1 2 √ k √ 4A2b2 1k + 4 A2b2 2k − 4b2 1k2k1 − 4b2 2k2k1 + k2), (13) and replacing k2 with k3, and bj with aj, yields the general solution of Eq. (6), i.e., w3 = ± ¿ Á ÁÀ2( √ 2Aky2 − k2y4 − kk1a1 + Aa2 − a2ky2 − 1 2 √ k √ 4A2a2 1k + 4 A2a2 2k − 4a2 1k2k1 − 4a2 2k2k1 + k3), (14) with a1, a2 being arbitrary constants. Thus, we have derived w2 and w3 as functions of y = w1, namely, the general solution of system (3), since p2 and p3 can be obtained by substituting w2 and w3 into the first and second equations in (3), respectively. Of course, integrating the first Hamiltonian equation in (2), i.e., ẇ1 = √ 2 w1 √ 2Aw2 1 − kw4 1 − k1, (15) will yield the general solution of the Hamiltonian equations (2). Therefore, we have shown that the maximally superintegrable Hamiltonian system (2) hides twice a third-order linear equation (9) and, consequently, that its general solution can be derived by substitutions and a final integration by quadratures. All of these have been accomplished without making use of any of the other four first integrals derived in Ref. 7 of which two are given in cartesian and two in spherical polar coordinates. However, we are more interested in showing that all the maximally superintegrable Hamiltonian systems hide linear equations, than in determining their general solutions. We observe that as the Hamiltonian H1 contains three copies of the one-dimensional caged oscillator, all variables wi lead to the same reduction. The procedure does not depend whether the frequencies are commensurable or not, indicating that the same reduction is valid for the case where the oscillator is not isotropic. This shows, in particular, that the superintegrable caged anisotropic oscillator with Hamiltonian, H(ℓ,m,n) 1 = 1 2 (p2 1 + p2 2 + p2 3) + k(ℓ2w2 1 +m2w2 2 + n2w2 3) + k1 w2 1 + k2 w2 2 + k3 w2 3 , (16) introduced in Ref. 11 also hides a third-order linear equation of the type (9), leading to the linearization of the system. In addition, as the Hamiltonian (16) always admits two independent integrals of orders 2(ℓ +m − 1) and 2(ℓ + n − 1) in the momenta, respectively, it follows that the reduction method is not dependent on the degree of the first integrals, or whether these arise from the separation of variables of the Hamilton–Jacobi equation, as it was shown in other such instances in Refs. 5 and 4. B. The potential V II (w 1, w 2, w 3) In Cartesian coordinates, the Hamiltonian is given by H(cc) 2 = 1 2 (p2 1 + p2 2 + p2 3) − k √ w2 1 +w 2 2 +w 2 3 + k1 w2 1 + k2 w2 2 . (17) As the variables wi in the potential are nontrivially coupled, it is convenient to reformulate the Hamiltonian in spherical coordinates in order to largely simplify the computations in the reduction. In spherical polar coordinates, we obtain H2 = 1 2 ⎛ ⎝ p2 r + p2 ϕ r2sin2(θ) + p2 θ r2 ⎞ ⎠ − k r + k1 r2sin2(θ)cos2(ϕ) + k2 r2sin2(θ)sin2(ϕ) , (18) J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-4 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp yielding the following Hamiltonian equations: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ṙ = pr , ϕ̇ = pϕ r2sin2(θ) , θ̇ = pθ r2 , ṗr = p2 ϕ r3sin2(θ) + p2 θ r3 − k r2 + 2k1 r3sin2(θ)cos2(ϕ) + 2k2 r3sin2(θ)sin2(ϕ) , ṗϕ= 2k2cos(ϕ) r2sin2(θ)sin3(ϕ) − 2k1sin(ϕ) r2sin2(θ)cos3(ϕ) , ṗθ = cos(θ) ⎛ ⎝ p2 ϕ r2sin3(θ) + k1 r2sin3(θ)cos2(ϕ) + k2 r2sin3(θ)sin2(ϕ) ⎞ ⎠ . (19) We apply again the reduction method.2 If we choose ϕ as a new independent variable y, then system (19) reduces to the following five equations: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ dr dy = r2sin2 (θ)pr pϕ , dθ dy = sin2 (θ)pθ pϕ , dpr dy = pϕ r + sin2 (θ)p2 θ rpϕ − ksin2 (θ) pϕ + 2k1 rcos2(y)pϕ + 2k2 rsin2(y)pϕ , dpϕ dy = 2k2cos(y) sin3(y)pϕ − 2k1sin(y) cos3(y)pϕ , dpθ dy = cot(θ) pϕ (p2 ϕ + k1 cos2(y) + k2 sin2(y) ). (20) The fourth equation of system (20) can be integrated to yield p2 ϕ 2 + k1 cos2(y) + k2 sin2(y) = A = const. Ô⇒ pϕ = ± √ 2 √ A − k1 cos2(y) − k2 sin2(y) . (21) We observe that this is identical to the first integral I2 given in Ref. 7. Deriving pθ from the second equation of system (20), i.e., pθ = √ 2 sin2(θ) dθ dy √ A − k1 cos2(y) − k2 sin2(y) , (22) and replacing it into the fifth equation lead to the following second-order equation in θ: d2θ dy2 = 2cot(θ)( dθ dy ) 2 + Acos2 (y)sin2 (y) Acos2(y)sin2(y) − k1sin2(y) − k2cos2(y) sin(θ)cos(θ) + k1sin4 (y) − k2cos4 (y) cos(y)sin(y)(Acos2(y)sin2(y) − k1sin2(y) − k2cos2(y)) dθ dy . (23) This equation admits an eight-dimensional Lie symmetry algebra and is therefore linearizable. One simple symmetry is −cos(θ)sin(θ)∂θ, which can be transformed into the normal form u∂u by the change of the dependent variable u = −cot(θ). Then, Eq. (23) becomes the following linear equation: u′′ = (k1sin(y)4 − k2cos(y)4 )u′ − Asin(y)3cos(y)3u sin(y)cos(y)(Asin(y)2cos(y)2 − k1sin(y)2 − k2cos(y)2) . (24) Now, deriving pr from the first equation of system (20) and replacing it into the third equation yield a second-order equation in r, the coefficients of which involve the general solution of Eq. (23). It is explicitly given by J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-5 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp d2r dy2 − 2 r ( dr dy ) 2 +F(y)dr dt + G(y)r +H(y)r2 = 0, (25) where F(y) = 2(k1 − k2)cos4 (y) − k1cos(2y) sin(2y)(Asin2(y)cos2(y) − k1sin2(y) − k2cos(2)y) − 2cot(θ) dθ dy , G(y) = sin2 (θ)sin2 (y)cos2 (y) Acos4(y) + (k2 − k1 − A)cos2(y) + k1 − ( dθ dy ) 2 , H(y) = ksin2 (θ)sin2 (y) 2Acos4(y) + 2(k2 − k1 − A)cos2(y) + 2k1 . Equation (25) admits an eight-dimensional Lie symmetry algebra sl(3,R), and therefore, it is linearizable. Indeed, it can be transformed into the following linear equation by means of the change of the dependent variable z = −r−1: d2z dy2 +F(y) dz dy − G(y)z +H(y) = 0. (26) The reduction to the canonical form w′′(s) = 0 can be obtained using a generalized Kummer–Liouville transformation z = P(y)w(s) +Q(y), s = s(y), (27) where ds dy = 2(Acos4 (y) + k1 + (k2 − k1 − A)cos2 (y)) sin2(θ)sin(2y)P2(y) (28) and P(y) and Q(y) are a solution of the homogeneous part of Eq. (26) and a particular solution, respectively.12 We conclude that the maximally superintegrable Hamiltonian system (19) is linearizable since it hides two second-order linear equations (24) and (26). C. The potential V III (w 1, w 2, w 3) The Hamiltonian H3 = 1 2 (p2 1 + p2 2 + p2 3) + k1w1 w2 2 √ w2 1 +w 2 2 + k2 w2 2 + k3 w2 3 (29) yields the Hamiltonian equations ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ẇ1= p1, ẇ2= p2, ẇ3= p3, ṗ1 = − k1 (w2 1 +w 2 2) 3/2 , ṗ2 = k1w1(2w2 1 + 3w2 2) w3 2(w 2 1 +w 2 2) 3/2 + 2k2 w3 2 , ṗ3 = 2k3 w3 3 . (30) If we derive p3 from the third equation of system (30) and replace it into the sixth equation, we obtain the following second-order equation in w3: ẅ3 = 2k3 w3 3 . (31) It admits a three-dimensional Lie symmetry algebra sl(2,R) generated by the following operators: J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-6 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp t2∂t + tw3∂w3 , 2t∂t +w3∂w3 , ∂t . (32) Solving Eq. (31) with respect to k3 and deriving it once with respect to t lead to the third-order equation, ... w3 = − 3ẇ3ẅ3 w3 , (33) which is easily seen to admit a seven-dimensional Lie symmetry algebra generated by the following operators: X1 = t2∂t + tw3∂w3 , X2 = t∂t , X3 = ∂t , X4 = w3∂w3 , X5 = t2 w3 ∂w3 , X6 = t w3 ∂w3 , X7 = 1 w3 ∂w3 and is therefore linearizable. We find that a two-dimensional non-Abelian intransitive subalgebra is that generated by X4 and X7, and following Lie’s classification,8 if we transform these operators into their canonical form, i.e., ∂u, u∂u, then we obtain that the new dependent variable u = w2 3/2 transforms equation (33) into the linear equation ...u = 0, (34) which, solved and replaced into Eq. (31), yields w3 = ± ¿ Á ÁÀA1t2 + A2t + A2 2 + 8k3 4A1 (35) with An(n = 1, 2) arbitrary constants. About the other four equations of system (30), we make a simplifying substitution, i.e., w2 = √ r2 2 −w 2 1 . (36) If we derive p1 from the first equation of system (30) and replace it into the fourth equation, we obtain the following second-order equation in w1: ẅ1 = − k1 r3 2 . (37) Similarly, deriving p2 from the second equation of system (30) and replacing it into the fifth equation yield the second-order equation in r2, r̈2 = w2 1 ṙ2 2 r2(r2 2 −w 2 1) − 2w1ẇ1 ṙ2 r2 2 −w 2 1 + ẇ2 1r3 2 + 2k1w1 + 2k2r2 r2 2(r 2 2 −w 2 1) . (38) The system of Eqs. (37) and (38) admits a three-dimensional Lie symmetry algebra sl(2,R) generated by the following operators: t2∂t + tw1∂w1 + tr2∂r2 , 2t∂t +w1∂w1 + r2∂r2 , ∂t . (39) If we solve system (37) and (38) with respect to k1 and k2 and derive once with respect to t, the following system of two separated third-order equations is obtained: ... w1 = − 3ẇ1ẅ1 w1 , (40) ...r2 = − 3ṙ2 r̈2 r2 , (41) namely, both w1 and r2 satisfy the same Eq. (33) as w3. As a consequence, the transformations u1 = w 2 1/2 and u2 = r2 2/2 take Eqs. (40) and (41) into the linear equations, ...u1 = 0, (42) ...u2 = 0. (43) J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-7 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp We conclude that the maximally superintegrable Hamiltonian system (30) hides (three times) the third-order linear equation (34). We remark that, as shown in Ref. 7, the Hamilton–Jacobi equation corresponding to the Hamiltonian (29) is not separable in Cartesian coordinates. In contrast to the previous Hamiltonians, in this case, the shortest and simplest ansatz to linearize the system is a direct approach, i.e., without replacing the independent variable t by a new one and reducing the number of equations, although the reduction method can also be applied (see the Appendix). D. The potential V IV (w 1, w 2, w 3) The Hamiltonian H4 = 1 2 (p2 1 + p2 2 + p2 3) + k1w1 w2 2 √ w2 1 +w 2 2 + k2 w2 2 + k3w3 (44) can also be treated directly as the case H3, which turns out to be the shortest way to linearize the system. Its linearization using the reduction method is given in the Appendix. The equations associated with the Hamiltonian are given by ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ẇ1= p1, ẇ2= p2, ẇ3= p3, ṗ1 = − k1 (w2 1 +w 2 2) 3/2 , ṗ2 = k1w1(2w2 1 + 3w2 2) w3 2(w 2 1 +w 2 2) 3/2 + 2k2 w3 2 , ṗ3 = −k3. (45) The last linear equation can be immediately integrated to give p3 = −k3t + A1, which replaced into the third equation of the Hamiltonian system (45) yields a linear equation in w3 whose solution is w3 = − k3 2 t2 + A1t + A2. The remaining four equations of system (45) are treated in exactly the same way as in the preceding case, starting with the simplifying substitution (36). Deriving p1 from the first equation of system (45) and replacing it into the fourth equation yield the second-order equa- tion (37), as well as deriving p2 from the second equation of system (45) and replacing it into the fifth equation lead to the second-order equation (38). It follows that the maximally superintegrable Hamiltonian system (45) hides (twice) the third-order linear equation (34). E. The potential V V (w 1, w 2, w 3) Although the Hamiltonian H5 = 1 2 (p2 1 + p2 2 + p2 3) + k(w2 1 +w 2 2) + 4kw2 3 + k1 w2 1 + k2 w2 2 (46) can be seen, as H1, as a three-dimensional extension of a plane system containing two copies of the one-dimensional caged oscillator, the simplest way to linearize the system is, again, the direct approach. For H5, we obtain the Hamiltonian equations, ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ẇ1= p1, ẇ2= p2, ẇ3= p3, ṗ1 = 2 k1 − kw4 1 w3 1 , ṗ2 = 2 k2 − kw4 2 w3 2 , ṗ3 = −8kw3. (47) J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-8 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp Deriving p3 from the third equation of system (47) and replacing it into the sixth equation yield the following linear second-order equation in w3: ẅ3 = −8kw3, (48) whose general solution is given by w3 = A1cos(2 √ 2kt) + A2sin(2 √ 2kt), (49) if we assume k > 0. Deriving p1 from the first equation of system (47) and replacing it into the fourth equation yield the following second-order equation in w1: ẅ1 = 2 k1 − kw4 1 w3 1 . (50) Deriving p2 from the second equation of system (47) and replacing it into the fifth equation yield the following second-order equation in w2: ẅ2 = 2 k2 − kw4 2 w3 2 , (51) which is equal to Eq. (50) if k2 is substituted with k1. Equation (50) admits a three-dimensional Lie symmetry algebra sl(2,R) generated by the following operators: ∂t , cos(2 √ 2kt)∂t − √ 2ksin(2 √ 2kt)w1∂w1 , sin(2 √ 2kt)∂t + √ 2kcos(2 √ 2kt)w1∂w1 . (52) If we solve Eq. (50) with respect to k1 and derive once with respect to t, then we obtain the following third-order equation: ... w1 = −ẇ1 3ẅ1 + 8kw1 w1 , (53) which admits a seven-dimensional Lie symmetry algebra generated by the following operators: Y1 = ∂t , Y2 = cos(2 √ 2kt)∂t − √ 2ksin(2 √ 2kt)w1∂w1 , Y3 = sin(2 √ 2kt)∂t + √ 2kcos(2 √ 2kt)w1∂w1 , Y4 = w1∂w1 , Y5 = cos(2 √ 2kt) w1 ∂w1 , Y6 = sin(2 √ 2kt) w1 ∂w1 , Y7 = 1 w1 ∂w1 and is therefore linearizable. A two-dimensional non-Abelian intransitive subalgebra is generated by Y4 and Y7. Bringing them into the canonical form u∂u, ∂u we have that the new dependent variable u = w2 1/2 transforms equation (53) into the linear equation, ...u = −8ku̇. (54) The case k < 0 is completely analogous and leads to the same result. We conclude that the maximally superintegrable Hamiltonian system (47) hides (twice) the third-order linear equation (54). We observe that system (47) is quite similar to system (2), as H5 contains two copies of a one-dimensional caged oscillator with the addition of an ordinary oscillator. Therefore, as both systems contain a common two-dimensional subsystem and merely differ in the exten- sion to three dimensions, it would have been conceivable to linearize system (47) by using the same reduction as for system (2), although the computations are more involved. Indeed, the second-order equation that can be obtained by considering w1 = y a new independent variable is identical to Eq. (5), while the resulting second-order equation in w3 would be d2w3 dy2 = − (k1 − ky4 ) k1 + ky5 − 2Ay3 dw3 dy − 4ky2 k1 + ky4 − 2Ay2 . (55) This linear equation admits the maximal symmetry sl(3,R) and thus can be reduced to the free form u′′ = 0 by a point transformation.8,9 Hence, we conclude that system (47) hides two linear equations of second- and third-order, respectively. J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-9 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp III. CONCLUDING REMARKS Using a combination of various techniques of Lie point symmetries, notably the classical Lie criterion addressing the linearization of a system that admits a Lie point symmetry algebra of maximal dimension,8,9 the reduction method for autonomous systems introduced in Ref. 2, as well as a method to linearize second-order ordinary differential equations with sl(2,R) symmetry,10 we have shown that all max- imally superintegrable systems on flat space classified in Refs. 7 and 11 admit hidden symmetries leading to linearization. For some of the potentials, two possible ways for linearization have been presented, a direct approach that does not involve a change of the independent vari- able, as well as the application of the reduction method of Ref. 2, which may result in more complicated equations and computations. It has been illustrated that the linearization process applied to each of the potential is independent on the separating coordinates of the system and does not rely on the degree of the first integrals. We recall that although in Ref. 7, the admitting integrals are at most quadratic in the canon- ical momenta, on the other end in Ref. 11, there are two independent integrals of higher orders. Consequently, it follows that the reduction method is not dependent on the degree of the first integrals or whether these arise from the separation of variables of the Hamilton–Jacobi equation, as it was shown in other such instances in Refs. 5 and 4. This may be a hint that other maximally superintegrable systems for which the first integrals are of higher-order may also hide some linear equations. Regardless of their particular degree, we have not used explicitly the fact that the number of independent first integrals is maximal, which leads us to ask if the procedure is also valid for Hamiltonian systems in three-dimensional space having less than five independent constants of the motion. The following example illustrates that a minimally superintegrable system can also be linearizable. To this extent, consider the Hamiltonian7 H = 1 2 (p2 1 + p2 2 + p2 3) + 4kw2 1 + kw2 2 + k2 w2 2 + F(w3), (56) where k, k2 are constants and F(w3) is an arbitrary function. The Hamiltonian system ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ẇ1= p1, ẇ2= p2, ẇ3= p3, ṗ1 = −8kw1, ṗ2 = −2kw2 + 2k2 w3 2 , ṗ3 = − dF(w3) dw3 (57) is minimally superintegrable with four globally defined first integrals, separable in both Cartesian and parabolic cylindrical coordinates.7 Applying the reduction method of Ref. 2 with w3 as new independent variable y, the system is reduced to the following five equations: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ dw1 dy = p1 p3 , dw2 dy = p2 p3 , dp1 dy = − 8kw1 p3 , dp2 dy = − 2kw2 p3 + 2k2 w3 2p3 , dp3 dy = − F′(y) p3 . (58) The last equation is easily integrable and provides p3 = ± √ A − 2F(y). (59) We take into consideration the positive root, without loss of generality. If we now derive p1 and p2 from the first and second equations of system (58) and insert them into the third and fourth equations, respectively, then we are led to the following two second-order equations: w′′1 (y) = F′(y)w′1(y) − 8kw1(y) A − 2F(y) (60) and J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-10 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp w′′2 (y) = F′(y)w′2(y)w 3 2(y) − 2kw4 2(y) + 2k2 w3 2(y)(A − 2F(y)) . (61) Equation (60) is linear and thus, as shown in Ref. 8, admits an eight-dimensional Lie symmetry algebra, i.e., sl(3,R). Equation (61) can be shown to admit a three-dimensional Lie symmetry algebra [unless k2 = 0, for Eq. (61) becomes linear], i.e., sl(2,R), generated by the following operators: 2S(y)∂y + (S′(y) + F′(y)S(y) A − 2F(y) )w2∂w2 , (62) where S(y) is the solution of the following third-order linear equation: (A − 2F)S′′′ + (2(A − 2 f )2F′′ + 3(A − 2F)F′2 + 8k(A − 2F)2 )S′ + ((A − 2F)2F′′′ + 5(A − 2F)F′F′′ + 6F′3 + 8k(A − 2F))S = 0. (63) If we solve Eq. (61) with respect to the constant k2, i.e., k2 = w3 2 2 ((A − 2F)w′′2 − F′w′2 + 2kw2), (64) and derive it with respect to y, the resulting third-order equation, i.e., w′′′2 = 1 (A − 2F)w2 (3(F′w2 − (A − 2F)w′2)w ′′ 2 + (3F′w′2 + (F ′′ − 8k)w2)w ′ 2), (65) admits a seven-dimensional Lie symmetry algebra, showing that it is linearizable. Indeed, if we make the transformation u = w2 2 , then Eq. (65) becomes the following linear equation: u′′′ = 1 (A − 2F) (3F′u′′ + (F′′ − 8k)u′). (66) We conclude that the Hamiltonian system (57) hides a second-order linear equation and a third-order linear equation, regardless of the arbitrary function F(w3). In this context, it constitutes a natural question whether the remaining equivalence classes of minimally superintegrable Hamiltonian systems classified in Ref. 7 are also linearizable. Work in this direction is currently in progress. An answer in the positive would indicate that the Lie symmetry method, a technique that has somehow been neglected in the context of superintegrable systems, is an approach potentially relevant to their analysis as the separability problem of the Hamilton–Jacobi equation. ACKNOWLEDGMENTS M.C.N. acknowledges the partial support from the University of Perugia through Fondi Ricerca di Base 2018. R.C.-S. was supported by the research project MTM2016-79422-P of the AEI/FEDER (EU). APPENDIX: LINEARIZATION OF THE HAMILTONIANS H3 AND H4 BY THE REDUCTION METHOD We show that the superintegrable systems with Hamiltonians H3 and H4 that have been treated directly, without explicitly reducing the system with respect to a new variable, can also be linearized using the reduction method. However, this procedure leads to more complicated computations than the direct approach. First of all, we observe that the Hamiltonians H3 and H4 can be seen as an extension of the two-dimensional Hamiltonian He = 1 2 (p2 1 + p2 2) + k1w1 w2 2 √ w2 1 +w 2 2 + k2 w2 2 . (A1) More precisely, we have J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-11 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp H3 = He + 1 2 p2 3 + k3 w2 3 , H4 = He + 1 2 p2 3 + k3w3. (A2) Then, we should begin by looking at the common two-dimensional Hamiltonian He and then analyzing the three-dimensional cases. In this case, the use of polar coordinates is best suited, and indeed, if we introduce them, i.e., w1 = r cos(ϕ), w2 = r sin(ϕ), then the Hamiltonian He is transformed as Hep = 1 2 (p2 r + pϕ r2 ) + k1cot(ϕ) sin(ϕ)r2 + k2 r2sin(ϕ)2 . (A3) This Hamiltonian is a particular case of a more general two-dimensional Hamiltonians Hep = 1 2 (p2 r + pϕ r2 ) + α r + 1 r2 F(ϕ) (A4) which was shown in Ref. 4 to hide a second-order linear equation for any function F(ϕ). The reduction is carried out considering ϕ as a new independent variable y from which it follows that [see Eq. (67) in Ref. 4 and the following discussion fora detailed discussion of this potential] dy dt = pϕ r2 , pϕ = ± √ A − 2F(ϕ). (A5) This same reduction can be applied to the extended three-dimensional case if we express the Hamiltonians H3 and H4 in cylindrical coordinates, w1 = rcos(ϕ), w2 = rsin(ϕ), w3 = z, so that the Hamiltonians are given by H3 = Hep + 1 2 p2 z + k3 z2 , H4 = Hep + 1 2 p2 z + k3z, (A6) respectively. Then, the equations for ż and ṗz with respect to the new independent variable ϕ = y become the following equations for the Hamiltonian H3: dz dy = r2pz pϕ , dpz dy = 2k3r2 pϕz3 , (A7) while those for H4 become dz dy = r2pz pϕ , dpz dy = − k3r2 pϕ . (A8) Deriving now pz from the first equation in (A7), inserting it into the second equation taking into account the expression for pϕ, and simplifying the resulting expression yield the following second-order equation for z in the case of the Hamiltonian H3: d2z dy2 = d dy ln ⎛ ⎝ r2sin(y) 3 2 √ Asin(y)3 − k1cos(y) − k2sin(y) ⎞ ⎠ dz dy + 2k3r4sin(y)3 (Asin(y)3 − k1cos(y) − k2sin(y))z3 , (A9) while for the Hamiltonian H4, it is d2z dy2 = d dy ln ⎛ ⎝ r2sin(y) 3 2 √ Asin(y)3 − k1cos(y) − k2sin(y) ⎞ ⎠ dz dy − k3r4sin(y)3 (Asin(y)3 − k1cos(y) − k2sin(y)) . (A10) Equation (A9) can be simplified by means of the additional change of the dependent variable, J. Math. Phys. 62, 012702 (2021); doi: 10.1063/5.0007377 62, 012702-12 Published under license by AIP Publishing https://scitation.org/journal/jmp Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp z = u rsin(y) 3 4 (Asin(y)3 − k1cos(y) − k2sin(y)) 1 4 , (A11) leading to the differential equation in u, d2u dy2 = T(y)u + 2k3 u3 , (A12) with T(y) expressed in terms of the known functions r and pϕ as T(y) = 2 r2 ( dr dy ) 2 − 1 r d2r dy2 + 1 2pϕ d2pϕ dy2 − 1 4p2 ϕ ( dpϕ dy ) 2 − 1 rpϕ dr dy dpϕ dy . (A13) Equation (A12) admits a symmetry algebra isomorphic to sl(2,R), since it is a Pinney equation (see, e.g., Ref. 10). Solving this equation with respect to k3 and deriving with respect to y lead to the third-order differential equation, d3u dy3 = − dT dy u − 4T(y)du dy − 3 u du dy d2u dy2 , (A14) which can be further reduced to a linear equation, regardless of the function T(y), by means of a change of dependent variable U = 1 2 u2, d3U dy3 = −2 dT dy U − 4T(y)dU dy . (A15) Consequently, the maximally superintegrable Hamiltonian system (30) hides a second-order linear equation [see Eq. (80) in Ref. 4] and a linear third-order equation (A15), while the Hamiltonian system (45) hides two linear second-order equations, i.e., Eq. (80) in Ref. 4 and Eq. (A10). Thus, we have shown that systems (30) and (45) hide linear equations by two different approaches. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1L. P. Eisenhart, “Enumeration of potentials for which one-particle Schrödinger equations are separable,” Phys. Rev. 74, 87–89 (1948). 2M. C. Nucci, “The complete Kepler group can be derived by Lie group analysis,” J. Math. Phys. 37, 1772–1775 (1996). 3M. C. Nucci and P. G. L. Leach, “The harmony in the Kepler and related problems,” J. Math. Phys. 42, 746–764 (2001). 4G. Gubbiotti and M. C. Nucci, “Are all classical superintegrable systems in two-dimensional space linearizable?,” J. Math. 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