Global integrability of cosmological scalar fields

Global in tegrabilit y of cosmolog ical scalar fields Andrzej J. Maciejews ki, Institute of Astronom y , Univ ersit y of Zielona G´ ora P o dg´ o rna 50, 65–246 Zielona G´ ora, Poland, (e- mail: maciejk a @astro.ia.uz.zgora.pl) Maria Przyb ylsk a , T oru ´ n Cen tre for Astronom y , Nicholaus Cop ernicus Univ ersity Gagarina 1 1, 87– 100 T oru ´ n, Poland, (e-mail: mprzyb@astri.uni.torun.pl) T omasz Stac how iak, Astronomical O bserv atory , Jagiellonia n Univers ity Orla 17 1, 30 -244 Krak´ ow, P ola nd, ( e-mail: toms@oa.uj.edu.pl) Marek Szyd lo wski, Marc Ka c Complex Systems Researc h Cen ter, Jagiellonian Unive rsity Reymon ta 4, 30-059 Krak´ ow, P oland and Astronomical O bserv atory , Jagiellonia n Univers ity Orla 17 1, 30 -244 Krak´ ow, P ola nd, ( e-mail: uoszydlo@cyf-kr.edu.pl) Octob er 2 4, 2018 Abstract W e inv estigate the Lio uvillian in tegrability of Hamiltonian systems descr ibing a universe filled with a scalar field (p ossibly complex). The to ol used is the differential Galois gr oup approach, a s intro duced b y Mor ales-Ruiz and Ramis. The main result is that the gener ic systems with minimal coupling are non-in tegr able, altho ugh ther e still e xist s ome v alues of parameters for which integrability remains undecided; the conformally coupled sys tems a re only integrable in four known c ases. W e also draw a connectio n with the chaos present in such cosmolog ical mo dels, and the is sues of the integrability r estricted to the real doma in. 1 In tro du c tion Homogeneous and isotropic cosmological mo dels, although v ery simple, explain the r ecen t observ ational data v ery well [ 57, 55]. Their foundation is the F riedmann-Rob ertson-W alk er (FR W ) un iv erse, describ ed by th e metric d s 2 = a ( η ) 2  − d η 2 + d r 2 1 − K r 2 + r 2 d 2 Ω 2  , (1) 1 where a is the scale f actor, d 2 Ω 2 is the line elemen t on a t wo-sphere, and w e c hose to u se the conformal time η . A s can b e seen from the ab o v e metric, the scale facto r represents the relativ e c hange in th e distance of tw o p oints whose spatial coord inates a r e fixed. It dep ends only o n time, so that the whole unive r se is deformed in a h omogeneous fashion. F rom the an throp o cen tric p oint of view it could b e seen as a three d imensional sp ace evolving in the time - in th e simplest case when the curv ature index K is zero, it would b e a Euclidean space stretc hed according to a . If we we re to fill s uc h a u niv erse with a matter, its prop erties could only d ep end on time, and b e the same in all p oin ts of th e spatial subspace at a give n v alue of η – otherwise the mo del w ould no longer b e homogeneous. F or example a p erfect fluid wo uld b e completely describ ed by t wo qu an tities – its density and pressure as functions of time. A scalar field w ould b e describ ed by its field v ariables. Also a cosmolog ical constan t with its trivial dep end ence on time could alwa ys b e includ ed in such mo dels. Dep ending on the matter comp onents one obtains v arious ev olutions of the scale factor a , as giv en by the ge n eral action I = c 4 16 π G Z  R − 2Λ − 1 2  ∇ α ¯ ψ ∇ α ψ + V ( ψ ) + ξ R| ψ | 2  −   √ − g d 4 x , (2) where R is the Ricc i scalar, Λ the cosmolo gical constant, V the field’s p oten tial, ξ the coupling constan t, and  is the densit y of the p erfect flu id. The p oten tial usually includes at least a quadratic term m 2 | ψ | 2 , where m is the so-called mass of the field. When ξ = 0 w e sa y that the field is minimally coupled – it do es not uncouple since the determinan t of the metric g m ultiplies the whole Lagrangia n densit y . Case with ξ = 1 6 is the so-ca lled conformal coupling. F or the considered geometry , the ab o v e action ca n b e simplified so that it allo ws the Hamilto- nian a p proac h with the p hase v ariables depen ding only on conformal time η . Du e to th e required co v ariance of general relati vity , the system is sub ject to constrain ts, w hic h in our case mean that the obtained Hamiltonian’s v alue is zero. Ho wev er, we note that includ ing an additional matter comp onent  is equiv alen t to considering other energy lev els. Namely for  ∝ a − 4 (whic h is the case for radiation) a constan t is added to the Hamiltonian, thus imitating its non-zero v alue. This is the j ustification for studying the sys tems integrabilit y on a generic energy hyp er-surf ace. F rom the observ ational p oin t of view, the cosmologic al constan t p ro vides an explanation for the current accelerat ing expans ion of the unive rs e (Λ Cold Dark Matter mo del [22 ]), but a b etter solution still is sought f or. A real s calar field d ubb ed “quin tessence” with the so-call ed slo w rolling p oten tial, whic h mo d els the dark energy comp onent has b een extensiv ely us ed for that pur p ose [14, 68]. Re alisations of the field itself includ e also Bose-Einstein condensate of axions [25] or a ph an tom violating the energy principle [15]. Finally , scalar field could also b e the m ec hanism b ehind the inflation [43, 42], w hic h is currentl y the most established and used s cenario f or the early Univ erse [13]. Recen tly , Komatsu et a l. [39] ha ve sho wn that latest observ ational data (WMAP , SNIa, Barion Oscillation P eak and others) show that the mo del of chaot ic infl ation (whic h strictly sp eaking sh ould b e calle d n on- in tegrable or complex in the sens e that we demonstr ate in the p resen t work) w ith the quadratic p oten tial remains a go o d fit (within the 95% confi dence domain). F rom the physical p oint of view, a u niv erse fi lled with only on e comp onen t seems simple enough but it is not the case here. Chaotic s cattering has b een foun d in m inimally coupled 2 fields [23], as h as c h aotic dynamics [48]. The first of our r esults is that minimally coupled fi elds are not int egrable in the generic case. There are ho wev er sp ecial families of the sy stem’s parameters which lea v e the qu estion op en. W e giv e the appr opriate cond itions in the concludin g section. There are sev eral physical reasons to stud y more than ju st minimally coupled fields. Early w orks on c haotic in flation found the coupling constant ξ sm all or negativ e [30] but some argue [26] that the paradigm of infl ation should b e generalised to th e case with non-zero coupling constan t which should not b e fine-tuned close to zero, and WMAP observ ations seem to indicate non-negligible ξ . The coup ling could b e generated b y quan tum corrections [1 0 , 29], or from the r enormalisation of the K lein-Gordon equation as describ ed in [16]. The couplin g constan t sh ould b e fi xed by particle physic s of the matter comp osing the s calar field, for examp le the w a y ξ = 1 / 6 w as found in the large N app ro ximation to the Nam bu -Jona-Lasimo mo d el in [34]. Non-minimally coupled fields are also int eresting in the con text of d escription of the dark energy for which the ratio b et ween the p ressure and the ener gy density is less than − 1. Su c h matter is called a phanto m matter, and cannot b e ac hieved by standard scalar fields [27]. Conformally coupled fields we re sub ject to more rigorous inte grability analysis, as opp osed to minimally coupled ones, thanks to the n atural form of their Hamiltonian. As will b e sho wn in the next section, the kinetic p art is of n atural form, alb eit indefin ite, and the p otenti al is p olynomial (in the case of real fields). Chaos has b een studied in suc h fi elds by means of Lyapuno v exp onents, p erturbativ e ap- proac h, breaking up of the KAM tori [17, 11]. Also the P ainlev ´ e p rop erty [35] wa s emplo yed as an indicator of the system’s integrabilit y . Ziglin pro ved that the system giv en by (20) is not meromorphically int egrable w hen Λ = λ = 0 and k = 1 [67]. His metho d s w ere also used b y Y oshid a to homogeneous p oten tials wh ic h is the case for the system wh en k = 0 [61, 62 , 63, 64]. Later, Y oshida’s r esults w ere sharp ened by Morales-Ruiz and Ramis [49], and u sed by the pr esen t authors in [44] to obtain coun table familie s of p ossibly int egrable cases with some restrictions on λ and Λ. Also recen tly , more conditions for integrabilit y ha ve b een giv en in [12], although only for a non-zero sp atial curv ature k and a generic v alue of energy , that is, when the particular solution is a non-degenerate elliptic fun ction defined on a n on-zero energy lev el. Our w ork s ho ws, that the conjecture of th at pap er is in fact correct – as sho wn in Section 4 – the conform al system is only integrable in t wo cases (with the ab ov e assu mptions). W e go further than that and sho w that for a generic energy v alue, a spatially flat ( k = 0) the un iv erse is only integrable in four cases. Also, the particular case of zero en ergy is analysed and new, simple conditions on the m o del parameters are foun d. Finally , we chec k that w hen E = k = 0, the problem remains open, as the necessary conditions for in tegrabilit y are fulfilled. When it comes to numerical s tudies of the problem, there are v arious resu lts, most notably a c haotic b eha viour [36], bu t also a fractal s tructure and chao tic scattering [59]. Ho wev er, it remains unclear whether the widely exercised complex rotation of th e v ariables c h anges th e system’s in tegrabilit y . Even for v ery simple systems it was sho wn [31] that there migh t exist smo oth in tegrals, wh ic h are not real- analytic. This question is esp ecially vital since o ur Universe clearly has r eal initial conditions and dynamics. The results obtained for the conformal co u pling are m uc h stronger than in the minimal one. 3 W e manage to sho w, that th e four cases with kno wn fi rst in tegrals are the only int egrable ones for the generic energy v alue. The Hamiltonian of b oth these systems has ind efinite kinetic energy part, and to cast it in to a p ositiv e-defin ite form a transition into imaginary v ariables is used. It has b een done for a conformally coupled field [36], but some authors, see e.g. [54], argue that there are physica l limitations which f orbid extending the solutions through singu larities such as a = 0, and an imaginary scale facto r seems ev en less realistic. W e wo uld lik e to stress that the complexification of the v ariables in our appr oac h is not to b e connected with the physica l evolutio n of the sys tem int o th e imaginary v alues. Th e b eha viour of differenti al equations in the complex d omain is a to ol th at allo ws for obtaining general results regarding its inte grabilit y (also real), as can b e seen on the example of the Painlev ´ e analysis [35]. Despite the fact, that systems of the considered t yp e were often called non-in tegrable, there w as seldom a rigorous pro of of that prop osition. Ho w ever, th e Liouvillian in tegrabilit y can b e studied successfully , as w e try to sho w in this article. The authors are aw are of one notable attempt at stud ying the problem in [21]; sadly , that pap er cont ains a serious mistake in the application of Y oshid a’s theorem. The metho d used requires rescaling b y the energy of the system, which the authors of [21] tak e to b e zero. Th us, their Theorem 3 (wh ic h con tradicts one of th e results pr esen ted here) is in fact false. The zero energy lev el usually requires a separate treat ment, which we also pr esen t here. The plan of the pap er is the follo wing. In the next s ection w e describ e the mentioned t wo cosmologi cal m o dels: of minimally coupled field and conformally coupled fi eld starting f rom a general description up to the Hamiltonian formulation of these mo dels. Section 3 is d ev oted to in tro du ction to the Morales-Ramis theory . This is our to ol to pro ve the announced rigorous in tegrabilit y resu lts for these mo dels. In the n ext t wo Sections 4 and 5 all details of in tegrabilit y analysis are present ed. F or con ve n ience of readers all integ rability r esults are r ecapitulated in Section 6. 2 Ph ysical sy stem’s setup 2.1 Minimally coupled field The general acti on (2) no w includes the follo wing parts I = c 4 16 π G Z  R − 2Λ − 1 2  ∇ α ¯ ψ ∇ α ψ + m 2 ~ 2 | ψ | 2  √ − g d 4 x , (3) Using the coordin ates of (1), the Lagrangian b ecomes L = 6(¨ aa + K a 2 ) + 1 2 | ˙ ψ | 2 a 2 − m 2 2 ~ 2 a 4 | ψ | 2 − 2Λ a 4 , (4) with the dot denoting the deriv ativ e with resp ect to time. W e also dropp ed a co efficien t which includes some ph ysical constan ts and the part of the act ion related to the spatial in tegration. 4 Next we s ubtract a full deriv ative 6( ˙ aa ) ˙ , and use th e p olar parametrisation for the scalar field ψ = √ 12 φ exp( iθ ) to get L = 6( K a 2 − ˙ a 2 ) + 6 a 2 ( ˙ φ 2 + φ 2 ˙ θ 2 ) − 6 m 2 ~ 2 a 4 φ 2 − 2Λ a 4 , (5) and obtain the Hamilt onian H = 1 24  1 a 2 p 2 φ − p 2 a + 1 a 2 φ 2 p 2 θ  − 6 K a 2 + 2Λ a 4 + 6 m 2 ~ 2 a 4 φ 2 , (6) with p a = − 12 ˙ a, p φ = 12 a 2 ˙ φ, p θ = 12 a 2 φ 2 ˙ θ . (7) Note that th e the elliptic constrain ts of general r elativit y requ ire that the ab o ve Hamiltonian is iden tically zero – this is th e so-called F riedmann equation, although it is n ot a dynamical ev olution equation but rather a conserv ation la w [47]. Since θ is a cyclical v ariable, the corresp onding momen tum is conserv ed so we su bstitute p 2 θ = 2 ω 2 . T o mak e all the quan tities d imensionless, w e mak e the follo wing rescaling m 2 → m 2 ~ 2 | K | , Λ → 3Λ | K | , ω 2 → 72 ω 2 | K | , p 2 a → 72 p 2 a | K | , p 2 φ → 72 p 2 φ | K | . (8) There is no need of c h anging the v ariables a and φ along with their momen ta, as this is r eally c hanging the time v ariable η , and th us th e deriv ativ es to whic h the momen ta are prop ortional. This results also in d ividing the whole Hamilt onian by 6 √ 2 | K | to yield √ 2 H = 1 2  − p 2 a + 1 a 2 p 2 φ  − K | K | a 2 + Λ a 4 + m 2 φ 2 a 4 + ω 2 a 2 φ 2 . (9) If the sp atial curv ature is zero, any of th e other dimensional constan ts can b e used for this purp ose, so without the loss of generalit y w e take the righ t-hand s ide to b e the new Hamiltonian H = 1 2  − p 2 a + 1 a 2 p 2 φ  − k a 2 + Λ a 4 + m 2 φ 2 a 4 + ω 2 a 2 φ 2 , (10) and in all physical cases, k ∈ {− 1 , 0 , 1 } , ω 2 > 0, m 2 ∈ R , Λ ∈ R , H = 0. W e extend th e analysis somewhat assuming that the Hamiltonian might b e equal to some non-zero constant E ∈ R . W e will late r see, that our analysis includes also th e p ossibilit y of these co efficien ts b eing complex. Note that for a massless ( m = 0) field, the system is already solv able, as shown in Ap - p endix A. F rom this p oin t on, w e tak e ω = 0, whic h means the ph ase is constan t. Since the mo del h as U(1) symmetry , w e can alw a ys mak e suc h a fi eld real with a r otation in the complex ψ plane. In other w ords, we will b e in vesti gating a real scalar field only . The reason wh y we restrict the problem is the follo wing: the m etho d emplo yed requires an explicit (non-constan t) particular solution, and the only one kn o wn requires φ = 0; whic h is a singularity of the full Hamiltonian. Under the ab o v e assump tion the Hamilto n ’s equations of system (10) are ˙ a = − p a , ˙ p a = 2 k a − 4 a 3 (Λ + m 2 φ 2 ) + 1 a 3 p 2 φ , ˙ φ = 1 a 2 p φ , ˙ p φ = − 2 m 2 a 4 φ. (11) 5 W e n ote th at there is an obvious particular solution, which describ es an empt y un iv erse: φ = p φ = 0, a = q , p a = − ˙ q . Thanks to the energy in tegral E = 1 2 ˙ q 2 + kq 2 − Λ q 4 , it can b e iden tified with an appropriate elliptic function. 2.2 Conformally coupled scalar fields The pro cedur e of obtaining th e Hamiltonian is the same as in th e case of minimally coupled fields, only this time the action is I = c 4 16 π G Z  R − 2Λ − 1 2  ∇ α ¯ ψ ∇ α ψ + m 2 ~ 2 | ψ | 2 + 1 6 R| ψ | 2  − λ 4! | ψ | 4  √ − g d 4 x , (12) where an additional coupling to gra vit y through th e Ricci scalar R , and a quartic p oten tial term with constan t λ are pr esen t, as opp osed to the minimal scenario. W e k eep the same n otation as b efore and express the in v olv ed quan tities in the same co ordinates and get L = 6(¨ aa + K a 2 ) − 1 2 ¨ aa | ψ | 2 + 1 2 | ˙ ψ | 2 a 2 − m 2 2 ~ 2 a 4 | ψ | 2 − λ 4! a 4 | ψ | 4 − 2Λ a 4 − 1 2 K a 2 | ψ | 2 , (13) from which w e r emo v e a full deriv ativ e, an d int ro duce new field v ariables ψ = √ 12 φ exp( iθ ) /a to obtain L = 6  ˙ φ 2 + φ 2 ˙ θ 2 − ˙ a 2 + K ( a 2 − φ 2 ) − m 2 ~ 2 a 2 φ 2 − Λ 3 a 4 − λφ 4  . (14) The associated Hamiltonian is H = 1 24  p 2 φ + 1 φ 2 p 2 θ − p 2 a  + 6  K ( φ 2 − a 2 ) + m 2 ~ 2 a 2 φ 2 + λφ 4 + Λ 3 a 4  , (15) with p a = − 12 ˙ a, p φ = 12 ˙ φ, p θ = 12 φ 2 ˙ θ . (16) W e can see th at θ is a cyclical v ariable b ecause we took th e p otent ial to dep end on the m o dulus of ψ only , so w e write a constan t instead of the resp ectiv e momentum p θ = ω . Finally , w e express ev erythin g in dimensionless qu an tities, rescaling the constan ts, bu t also the time and momen ta (as th ey are in fact time deriv ativ es), which results in rescaling the whole Hamiltonian. W e do this as follo ws m 2 → m 2 ~ 2 | K | , Λ → 3 2 Λ | K | , λ → 1 2 λ | K | , p 2 x → 144 p 2 x | K | , H → 1 12 p | K | H , (17) when K 6 = 0, and using another of th e dimensional constan ts otherwise. Th u s, eliminating a m ultiplicativ e constan t, the Hamiltonian reads H = 1 2  p 2 φ − p 2 a  + 1 2  k ( φ 2 − a 2 ) + ω 2 φ 2 + m 2 a 2 φ 2  + 1 4  Λ a 4 + λφ 4  , (18) with k ∈ {− 1 , 0 , 1 } ( K = k | K | ); ω , λ , Λ, m 2 ∈ R , and H = 0 in any physicall y p ossible setup. Exactly as in the previous case, th e zero v alue of the energy is a consequence of the constrain ts in tro d uced by general relativit y . 6 W e note that for m = 0 the system decouples, and is trivially in tegrable as sh o wn in Ap- p endix B. That is why we will assume m 6 = 0 henceforth. W e will also take ω = 0, that is, consider a scalar field equiv alen t to a real field after a unitary rotation in the complex ψ plane. W e c hange the field v ariables into the standard q and p ones for further computation, taking a = q 1 , p a = p 1 , φ = q 2 , p φ = p 2 . (19) The Hamilt onian is then H = 1 2  − p 2 1 + p 2 2  + V , V = 1 2  k ( − q 2 1 + q 2 2 ) + m 2 q 2 1 q 2 2  + 1 4  Λ q 4 1 + λq 4 2  . (20) 3 Differen tial G alois obstructions to in tegrabilit y Let ( M , ω ) b e a 2 n -dimens ional complex analytic symplectic manifold. F or a meromorph ic function H : M → C , w e denote b y V H the Hamiltonian v ector field generated by H and let us consider Hamiltonian equations d x d t = v H ( x ) , t ∈ C , x ∈ M . (21) W e assume that a non-constan t particular solution ϕ ( t ) of system (21) is kn o wn. I ts m aximal analytic con tinuation defines a Riemann surface Γ with the local co ordinate t . Linearisation of (21) around ϕ ( t ) yields v ariational equations of the follo w ing form ˙ ξ = A ( t ) ξ , A ( t ) = ∂ v H ∂ x ( ϕ ( t )) , ξ ∈ T Γ M . (22) Thanks to Hamiltonian characte r of the system the dim ension of v ariational equations can b e red uced by t wo. First w e use the fact th at a Hamiltonian system has at least one first in tegral namely Hamiltonian H , th us we can restrict system (21) to the manifold M ε = { x ∈ M | H ( x ) = ε } , where ε = H ( ϕ ( t )). Then w e consider the induced system on the normal bun dle N := T Γ M ε /T Γ of Γ ˙ η = e A ( t ) η , e A ( t ) η = π ⋆ ( T ( v )( π − 1 ξ )) , η ∈ N . (23) Here π : T Γ M ε → N is the pro jection. The system of 2 n − 2 equations obtained in this wa y is called the normal v ariational equations. W e can consider the entries of matrices A and e A as elemen ts of fi eld K := M (Γ) of mero- morphic f unctions on Γ. This field w ith differen tiation with resp ect to t as a deriv ation is a differen tial field. O nly co n stan t fun ctions from K ha v e a v anishin g deriv ativ e, so the su bfield of constan ts of K is C . It is ob vious that solutions of (22) are not n ecessarily elemen ts of K n . The fund amen tal theorem of the d ifferen tial Galois theory guarantees that there exists a differentia l field F ⊃ K 7 suc h that it conta ins n linearly in dep end en t (o ver C ) solutions of (22). The smallest differential extension F ⊃ K w ith this p rop erty is called th e Picard-V essiot extension. A group G of differen tial automorphisms of F which do es n ot change K is calle d the differential Galois group of equation (2 2 ). It c an b e sho w n that G is a linear algebraic g rou p. Thus, it is a u nion of disjoin t connected comp onents. One of them conta inin g the iden tity is called the identit y comp onent of G . Differen tial Galois theory was created as a to ol to ans w er the qu estion: whether a give n system of linear equations p ossesses a solution th at can b e w ritten in a closed form, i.e. is it solv able? The main theorem of this theory states that the necessary condition of solv abilit y in the class of Liouvillian fu nctions (i.e. by generalised quadratures) is solv abilit y of its d ifferen tial Galois group . W e can try to connect the integ r abilit y of th e original nonlin ear system with solv abilit y of its v ariational equations. Ho wev er, there is a more direct connection. Namely in eigh ties of XX cent u ry Z iglin observe d that if system (21) h as k ≥ 2 fu nctionally indep en den t meromorphic fir st integ r als, then v ariational equations (22) and also normal v ariational equations (23) p ossess k rational fi rst inte grals and moreo ve r the m ono dromy group (that is a subgroup of differen tial Galois group) has the same n umb er of inv arian ts [65, 66]. F ourteen yea rs later the relation b etw een first in tegrals and in v arian ts of th e differential Galois group w as analysed by Baider, Ch ur c hill, Ro d and S inger in [19]. Ho wev er the final f orm ulation of relations b etw een in tegrabilit y of Hamiltonian systems a n d prop erties of the differen tial Galois group o f v ariational equations due to Morales and Ramis [51, 49] w here in their analysis not only the pr esence of first in tegrals is tak en in to accoun t but also the consequences of th e inv olution of firs t in tegrals. Their main theorem that will b e the cru cial tool of our analysis is the follo win g. Theorem 1 (Morales-Ruiz and Ramis [49]) Assume that a Hamiltonian system is mer o- morphic al ly inte gr able in the Liouvil le sense in a neighb ourho o d of a phas e curve Γ and irr e gular singularities of the variational e quations along Γ do not c orr esp ond to phase p oints at the in- finity. Then the i dentity c omp onent of the differ ential Galois gr oup of the (normal) v ariational e quations asso ciate d with Γ is Ab elian. Let us explain assumptions concerning v ariational equatio n s in the ab ov e theorem. Usually the Riemann su rface corresp ond ing to the phase curve Γ is n ot compact so we compactify it adding some p oin ts. Typica lly these p oin ts corresp ond to equilibria or infi nite p oint s. In the later case we ha ve to add these p oint s to the phase space, i.e, we hav e to extend our original system in to a ‘bigger’ phase space. F or the extended system the requiremen t that the considered first in tegrals are meromorphic in a neigh b ourho o d of the phase cur v e put strong restrictions: they ha ve to b e meromorph ic at th e infinity . Thus if w e remo v e the assum ption ab out irregular singular p oints w e ha ve to r estrict the class of first integral s. Belo w we give a version of th e Morales-Ramis theorem without assu mptions concerning the regularit y of v ariational equations, whic h is adapted to Hamiltonian systems considered in th is pap er. Theorem 2 (Morales-Ruiz and Ramis [49]) Assume that a Hamilto nian system define d i n a line ar symple ctic sp ac e is gener ate d by a r ational Hamiltonian function and is r ational ly in- te gr able in the Liouvil le sense. Then the identity c omp onent of the differ ential Galois gr oup of the (normal) variational e quations asso ciate d with Γ is Ab elian. 8 In app lications the most difficu lt part is to c hec k the ab elianit y of v ariational equations. F ortunately , thanks to th e separation of v ariational equations in to t wo parts w e can restrict to its normal part and in this wa y to redu ce the dimension of the s ystem. F urtherm ore, b ecause the ab elian differen tial Galois group imp lies in particular that this group is solv able, thus w e can use directly all solv abilit y r esults concernin g some kn o wn equations such as e.g. hyp ergeometric equation, Whittak er equation, Lam´ e equations. In addition, for a linear second order equation with r ational coefficien ts there exists the closed algorithm, so-called Ko v acic algorithm [40], that decides whether equation is solv able in a class of Liouvillian function, yields explicit forms of solutions as w ell as determines the different ial Galois group . Th is is ac hieve d by pro vid ing necessary conditions for solv abilit y of the appropriate Galois group . The equations in question ha ve as their Galois group an algebraic subgroup of SL 2 ( C ), and s ince there are only th ree p ossibilities of those having a solv able identit y comp onen t, the pro cedure is arranged in three cases. They consist of analysing the equation’s singular p oin ts an d find ing an appr opriate p olynomial and an alge b raic f unction of p ossible degrees 2, 4 , 6, or 12; used to construct s olutions. This m eans that Theorem 1 yields r eally an effectiv e to ol for p ro ving the non-integrabilit y and distinguishing the cases susp ected ab out in tegrabilit y in the case when Hamiltonian dep end s on some ph ysical parameters, for examples of ap plications see references in [52 ]. It can happ en that a considered system satisfies all conditions of the ab o v e theorem, but nev ertheless it is not in tegrable. It is n othing strange as this theorem giv es only necessary conditions f or the inte grability , Th is shows a n eed of stronger necessary conditions for the in tegrabilit y . They were deve lop ed by C. S im´ o, J.J Morales and J.-P . Ramis [49, 50, 52] and are based on higher ord er v ariational equations (HVE’s). Theorem 3 Assume that a Hamiltonian system is mer omorphic al ly inte gr able in the Liouvil le sense in a neighb ourho o d of the analytic phase curve Γ , and the infinity is a r e gular sing u lar p oint of the variational e quations along Γ . Then the identity c omp onent of the differ ential Galois gr oup of k -th v ariational e quations along Γ is Ab elian for al l k ≥ 1 . F or v ariational equations (VE’s) of degree greater than one ther e is no more the s plitting of v ariational equations in to t wo parts: normal (NVE’s) and tangentia l (TVE’s) and there is n o reduction of system’s dimension and the analysis of th e differential Galois group of the whole system of v ariational equations is more in volv ed. F ortunately , in the case when v ariational equations are th e pr o duct of Lam ´ e equations with Lam ´ e-Hermite solutions Morales-Ruiz p ro ved in [49, 50, 52] that th e absence of logarithmic terms in solutions of h igher order v ariational equations is a necessary condition of ab elianit y of the identit y comp onent of their d ifferen tial Galois groups. The intereste d reader more d etailed and complete presenta tion of Morales-Ramis theory can find in [19 , 49, 51, 65 , 66] and of differentia l Galois theory in [8, 58, 37, 49, 56]. 9 4 Analysis of the minimally coupled fi eld 4.1 Λ = 0 case The system no w has the follo wing form ˙ a = − p a , ˙ p a = 2 k a − 4 m 2 a 3 φ 2 + 1 a 3 p 2 φ , ˙ φ = 1 a 2 p φ , ˙ p φ = − 2 m 2 a 4 φ. (24) Using the aforementio ned particular solution, for whic h the constan t energy condition b ecomes E = 1 2 ˙ q 2 + k q 2 , w e hav e as the v ariational equ ations      ˙ a (1) ˙ p (1) a ˙ φ (1) ˙ p (1) φ      =     0 − 1 0 0 2 k 0 0 0 0 0 0 q − 2 0 0 − 2 m 2 q 4 0          a (1) p (1) a φ (1) p (1) φ      . (25) The n ormal part of the ab o ve system, after eliminating the momen tu m v ariation p (1) φ , and writing x for φ (1) , is q ¨ x + 2 ˙ q ˙ x + 2 m 2 q 3 x = 0 , (26) whic h we further simplify lik e b efore b y taking z = q as the new indep endent v ariable, and using the energy condition to get z ( E − k z 2 ) x ′′ + (2 E − 3 k z 2 ) x ′ + m 2 z 3 x = 0 . (27) W e c heck the physic al hyp ersur face of E = 0. Th is requir es k 6 = 0 f or otherw ise the sp ecial solution w ould b ecome an equilibrium p oin t. Intro ducing a n ew pair of v ariables w ( s ) = w  2 m √ k z  = z 3 / 2 x ( z ) , (28) w e finally get d 2 w d s 2 =  1 4 − κ s + 4 µ 2 − 1 4 s 2  w, (29) with µ = ± 1, and κ = 0. This is the Whittak er equ ation, and its solutions are Liouvillian if, and only if,  κ + µ − 1 2 , κ − µ − 1 2  are integ ers, one of them b eing p ositive an d the other negativ e [49]. As this is not the case here, this finishes the p ro of for k 6 = 0. W e recall, that b ecause of the irregular singular p oin t s = ∞ , this rules out only the rational fi rst in tegrals. Non-in tegrabilit y on one energy lev el m eans no global in tegrabilit y , for the existence of another in tegral for all v alues of E wo u ld imply its existence on E = 0. Ho wev er, there might exist add itional integrals for only some, sp ecial v alues of the energy . It is straightforw ard to c hec k w ith the u se of Kov acic’s algorithm [40], that th is is n ot tru e h ere. F or our equation, in cases 1 and 2 of the algorithm, there is n o app ropriate integ er degree of a p olynomial needed f or 10 the construction of the solution, and case 3 cannot h old, b ecause of the orders of the singular p oints of the equation. If k = 0, a c h ange of the dep endent v ariable to w ( z ) = z x ( z ), redu ces equatio n (27) to E w ′′ + m 2 z 2 w = 0 , (30) whic h is kno wn not to p ossess Liouvillian solutions [40]. W e notice th at when Λ = E = k = 0, the system can b e redu ced to a t wo -dimen sional one. In fact, the red uction is still p ossible when Λ 6 = 0, so we c h o ose to present in the next sectio n . 4.2 Λ 6 = 0 case W e use the nonzero constant Λ to rescale the sys tem as follo ws a = q 1 √ Λ , p a = p 1 √ Λ , φ = q 2 , p φ = p 2 Λ , (31) so that th e equations b ecome ˙ q 1 = − p 1 , ˙ p 1 = 2 k q 1 − 4 q 3 1 (1 + bq 2 2 ) + 1 q 3 1 p 2 2 , ˙ q 2 = 1 q 1 2 p 2 , ˙ p 2 = − 2 bq 2 q 4 1 , (32) where b = m 2 / Λ. The energy integral, for the p reviously defined particular solution, no w reads E = E Λ = 1 2 ˙ q 2 + k q 2 − q 4 , where q has b een r escaled according to (31). As b efore, w e are in terested in the v ariational equations, which read      ˙ q (1) 1 ˙ p (1) 1 ˙ q (1) 2 ˙ p (1) 2      =     0 − 1 0 0 2( k − 6 q 2 ) 0 0 0 0 0 0 q − 2 0 0 − 2 bq 4 0          q (1) 1 p (1) 1 q (1) 2 p (1) 2      , (33) and writing x f or q (1) 2 , and y for p (1) 2 . The normal part is ˙ x = 1 q 2 y , ˙ y = − 2 bq 4 x, (34) or alte r nativ ely ¨ x + 2 ˙ q q ˙ x + 2 bq 2 x = 0 . (35) 11 4.2.1 E = 0 W e first p ic k the p articular solution lying on the zero-energy lev el, as the global in tegra- bilit y implies the integ r abilit y for this particular v alue of the Hamiltonian. It is imp ortan t to remem b er, ho w eve r, th at the con v erse is not true. The normal v ariational equation is cast in to a rational form b y changing the indep enden t v ariable to z = q 2 /k (for k 6 = 0 w hic h implies k 2 = 1), and using the energy fir st integral . It then b ecomes a h yp ergeometric equation x ′′ + 5 z − 4 2 z ( z − 1) x ′ + b 4 z ( z − 1) x = 0 , (36) with the resp ectiv e c haracteristic exp onent s z = 0 , ρ = − 1 , 0 z = 1 , ρ = 0 , 1 2 z = ∞ , ρ = 1 4 (3 − √ 9 − 4 b ) , 1 4 (3 + √ 9 − 4 b ) . (37) By Kimura’s theorem [38], the solutions of equation (36) are Liouvillian if, and only if 9 − 4 b = (2 p − 1) 2 , p ∈ Z . As b efore, th is means that f or the global in tegrabilit y this condition must b e satisfied. F or k = 0 the solution of NVE is x 1 , 2 = q − 2 ρ ∞ 1 , 2 , and the reduction to a t wo -dimensional system is p ossible, as men tioned b efore. 4.2.2 E 6 = 0 The sp ecial solution, is n o w directly connected to the W eierstrass ℘ fu nction, for if w e in tro d uce a new dep enden t v ariable v w ith q 2 = 1 2 v + k 3 , (38) the energy in tegral implies that it satisfies the equatio n ˙ v 2 = 4 v 3 − g 2 v − g 3 , (39) where g 2 = 16 3 ( k 2 − 3 E ) , g 3 = 32 27 k (2 k 2 − 9 E ) , (40) and the discriminant ∆ = 1024 E 2 ( k 2 − 4 E ), which we tak e as n on-zero to consider the generic case. Th us, taking w = q (1) 2 q , and eliminating p (1) 2 as b efore, the n ormal v ariational equation reads ¨ w = [ A℘ ( η ; g 2 , g 3 ) + B ] w, (41) with A = 2 − b and B = − 2 3 k (1 + b ). Th is is the Lam ´ e differentia l equation, whose Liouvil- lian solutions are kno wn to fall into thr ee m utually exclusive cases, which are exactly those of Ko v acic’s algorithm: 12 1. Th e L am ´ e-Hermite case, with A = n ( n + 1) = 2 − b , n ∈ N . T his implies that 9 − 4 b = (2 n + 1) 2 . Case with n = 1 already kno w n to b e inte grable b ecause b = 0 represents the massless field. 2. Th e Briosc hi-Halphen-Cr a wford case, wh ere necessarily n is half an integ er, i.e. n + 1 2 = l ∈ N , and as b efore 9 − 4 b = (2 n + 1) 2 = (2 l ) 2 . 3. Th e Baldassarri case, with n + 1 2 ∈ 1 3 Z ∪ 1 4 Z ∪ 1 5 Z \ Z , and add itional algebraic r estrictions on B , g 2 , and g 3 . In the L am ´ e-Hermite case we h a v e infinite num b er of v alues of b = 2 − n ( n + 1), n ∈ N , for whic h the necessary conditions for the in tegrabilit y giv en by the Morale s-Ramis T heorem 1 are satisfied. In order to obtain stronger result w e need to apply a more restrictiv e necessary conditions. S uc h conditions are given b y higher order v ariational equations, see [52] for detailed exp osition. Here w e explain this tec h nique on the considered problem and w e will follo w [45]. A t the b eginning, it is con ve nient to change the v ariables in equations (32) in the follo wing w a y q 1 = w 1 , p 1 = − w 2 , q 2 = w 3 w 1 , p 2 = w 1 w 4 − w 2 w 3 . (42) Let ˙ w = W ( w ) , w = ( w 1 , w 2 , w 3 , w 4 ) , (43) b e the system (32) w ritten in the new v ariables. The adv an tage of new co ord inates is that no w the v ariational equations split in to a direct pro du ct of tw o Lam´ e equations      ˙ w (1) 1 ˙ w (1) 2 ˙ w (1) 3 ˙ w (1) 4      =     0 1 0 0 A 1 ℘ ( η ) + B 1 0 0 0 0 0 0 1 0 0 A 2 ℘ ( η ) + B 2 0          w (1) 1 w (1) 2 w (1) 3 w (1) 4      , (44) where ℘ ( η ) is the one give n by equation (39), and A 1 = 6 , B 1 = 2 k , A 2 = n ( n + 1) , B 2 = 2 3 k ( n 2 + n − 3) . (45) T o deriv e the higher order VE’s w e substitute in to equ ation (43) the infi nite formal series w = ϕ ( η ) + ǫw (1) + ǫ 2 w (2) + ǫ 3 w (3) + · · · , (46) where ϕ is the particular solution, and get ˙ w ( j ) = W ′ ( ϕ ( η )) w ( j ) + f j ( w (1) , . . . , w ( j − 1) ) , j = 1 , 2 , . . . , (47) 13 where W ′ ( ϕ ( η )) is the matrix of right hand sid es in (44), and f j ( w (1) , . . . , w ( j − 1) ) are v ectors obtained from the T a ylor expansions of comp onent s of W ( w ). In particular we h a v e f 1 = 0 , f 2 = 1 2 W ′′ ( ϕ ( η ))( w (1) , w (1) ) , f 3 = 1 6 W ′′′ ( ϕ ( η ))( w (1) , w (1) , w (1) ) + W ′′ ( ϕ ( η ))( w (2) , w (1) ) , (48) and so on. F or j = 1 equation (44 ) is reco v ered. Although w (1) , . . . , w ( j − 1) en ter p olynomially in the righ t hand sid es of j -th v ariational equations (47), there exists an app ropriate framewo rk to define their differen tial Galois group. In [52] it was pro ved that if the system is integ r able, then the ident ity comp onent G ◦ j of the differentia l Galois group G j of j -th v ariational equations is Ab elian. Generally it is v ery difficult to d etermine G j for j > 1. Ho w ev er, in a case when the first v ariational equations are a pro duct of t w o Lam´ e equations ha ving in finite differen tial Galois group w e ha ve an effectiv e metho d to decide whether G 0 j is Ab elian. Namely , if a logarithmic therm app ears in lo cal solution around η = 0 of j -th v ariational equ ations, then G 0 j is not Ab elian, see [52, 53] for details. The calculati ons pro ceed as follo ws. The solution of (47) is giv en by w ( j ) = X Z X − 1 f j d η , (49) where f j = f j ( w (1) , . . . , w ( j − 1) ) and X is the fun damen tal matrix of the homogeneous s ystem (i.e. the fi rst order VE (44)), so that ˙ X = W ′ ( ϕ ( t )) X, det X 6 = 0 . (50) W e to ok X =     v 1 v 2 0 0 ˙ v 1 ˙ v 2 0 0 0 0 v 3 v 4 0 0 ˙ v 3 ˙ v 4     , (51) with v 1 = η 3 + k 7 η 5 + · · · , v 2 = − 1 5 η 2 + k 15 + · · · , v 3 = η n +1 + k ( n 2 + n − 3) 6 n + 9 η n +3 + · · · , v 4 = − 1 (2 n + 1) η n + k ( n 2 + n − 3) (2 n + 1)(6 n − 3) η n − 2 + · · · . (52) Next, w e tak e as the solution of the first order VE w (1) = (0 , 0 , v 4 , ˙ v 4 ) . (53) Then we fix n = 2 and solv e the second order VE and w e obtain the in tegrand of (49) for j = 3 to b e X − 1 f 3 = (0 , 0 , 54 625 η 8 − 44 k 625 η 6 + · · · , 54 125 η 3 − 128 k 875 η + · · · ) , (54) 14 whic h pro d uces a logarithm in w (3) . If k = 0, one h as to find s olutions of fourth order VE to get X − 1 f 5 = (0 , 0 , − 3618 10937 5 η 10 − 1272 E 21875 η 6 + · · · , − 3618 21875 η 5 − 1536 E 21875 η + · · · ) , (55) whic h prov es the non-inte grability , sin ce we assumed E 6 = 0. This b eha viour do es not change as we increase n , although it was c hec ked only for 10 con- secutiv e v alues. W e thus c onje ctur e that for b = 2 − n ( n + 1) with in teger n > 1 the system is not int egrable. The pro cedur e describ ed is correct un der assumption that the differentia l Gal ois group of the normal v ariational equations is not finite. W e discuss th is p oin t in App endix C, and justify that except countable man y v alues of energy the group is not finite. In the Brioschi case, there is another add itional condition for the in tegrabilit y: the so-called Briosc hi determinan t Q l is zero [49]. Unfortun ately , th ere is no closed f orm ula for Q l for general l , but analysing the first few v alues w e notice a pattern: Q 1 = − 3 2 k , Q 2 = − 3 4 (5 k 2 − 16 E ) , Q 3 = − 9 8 k (35 k 2 − 192 E ) , Q 4 = − 5 16 (2835 k 4 − 21600 k 2 E − 4838 4 E 2 ) , Q 5 = − 4725 32 k (231 k 4 − 2240 k 2 E − 16384 E 2 ) , Q 6 = − 8505 64 (1501 5 k 6 − 176400 k 4 E − 2802432 k 2 E 2 − 1126400 E 3 ) . (56) When k = 0, Q l is zero for odd l , and prop ortional to energy , w hic h is not zero, for ev en l . When k 6 = 0, so that k 2 = 1, eac h Q l is a p olynomial in E , and th at giv es at most a finite n umb er of energy v alues f or which Q l = 0 and the system is p oten tially in tegrable. W e, again, c onje ctur e that if th e system is in tegrable (with this s ubsection’s assumptions) and k = 0, then necessarily n + 1 2 is od d, and that if k 2 = 1, then it is n ot in tegrable on a generic energy lev el. The Baldassarri case can also b e studied in more detail b y m eans of the mo dular function j = g 3 2 g 3 2 − 27 g 2 3 = 4( k 2 − 3 E ) 3 27( k 2 − 4 E ) E 2 = ( 4(1 − 3 E ) 3 27(1 − 4 E ) E 2 for k 2 = 1 , 1 for k = 0 . (57) A theorem by Dwork [49] states that the n umb er of pairs ( j, B ) is at most finite in integrable cases. Since j d ep ends on the energy for non-zero k , and B dep ends on m 2 , it means that a generic energy level is not in tegrable for a giv en v alue of m 2 . 4.2.3 E = k = 0 As men tioned in S ection 4.1 in this case we can transform the system to a t wo-dimensional one. In order to do th at, time needs to b e c h anged from the conformal to the cosmolo gical one 15 d η → d t = a d η in the original equations (11 ). W e then tak e as the new momen ta the Hubb le’s function a and the deriv ativ e of φ h := 1 a d a d t = − p a a 2 , ω := d φ d t = p φ a 3 . (58) (This ω is not to b e confu sed with the one introd uced in Section 2.) Accordingly we ha ve d a d t = ah, d φ d t = ω , d h d t = 4Λ + 4 m 2 φ 2 − ω 2 − 2 h 2 , d ω d t = − 2 m 2 φ − 3 ω h. (59) Th u s, w e are left with a dynamical s ystem in the ( h, φ, ω ) sp ace, as a decouples. F urth ermore, the energy in tegral is no w 0 = 1 2 a 4 (2Λ + ω 2 + 2 m 2 φ 2 − h 2 ) , (60) so for a ( t ) whic h is not tr ivially zero, it gives a first in tegral on the reduced space. Cho osing an appropriate v ariable α , suggested b y the form of this in tegral φ = √ h 2 − 2Λ √ 2 m sin( α ) , ω = p h 2 − 2Λ cos( α ) , (61) w e finally obtain d α d t = √ 2 m + 3 h sin ( α ) cos( α ) , d h d t = − 3( h 2 − 2Λ) cos 2 ( α ) . (62) The problem of such reduction w as also d iscussed in [28]. It is argued that there can b e no c haos in th is system, but its int egrabilit y – which would b e one more first integral – remains unresolv ed. 5 Analysis of the con formally coupled field 5.1 Kno wn integrable families There are four kno wn cases when the system has an additional first integ ral, fu nctionally indep end en t of th e Hamiltonian. They were found b y applying the so-calle d ARS algorithm basing on th e P ainlev ´ e an alysis [1]. T able 1 summarises those r esults. 16 case k Λ m 2 (1) 0 , ± 1 Λ = λ m 2 = − 3Λ (2) 0 , ± 1 Λ = λ m 2 = − Λ (3) 0 Λ = 16 λ m 2 = − 6 λ (4) 0 Λ = 8 λ m 2 = − 3 λ T able 1: Kno w n in tegrable cases for the conformally coupled field And the resp ectiv e in tegrals of the systems are (1)      H = 1 2 ( p 2 2 − p 2 1 ) + k 2 ( q 2 2 − q 2 1 ) − m 2 12 ( q 4 1 − 6 q 2 1 q 2 2 + q 4 2 ) , I = p 1 p 2 + 1 3 ( m 2 ( q 2 2 − q 2 1 ) − 3 k ) , (2)    H = 1 2 ( p 2 2 − p 2 1 ) + k 2 ( q 2 2 − q 2 1 ) − m 2 4 ( q 2 2 − q 2 1 ) 2 , I = q 1 p 2 + q 2 p 1 , (3)      H = 1 2 ( p 2 2 − p 2 1 ) − m 2 24 (16 q 4 1 − 12 q 2 1 q 2 2 + q 4 2 ) , I = ( q 1 p 2 + q 2 p 1 ) p 2 + m 2 6 q 1 q 2 2 ( q 2 2 − 2 q 2 1 ) , (4)      H = 1 2 ( p 2 2 − p 2 1 ) − m 2 12 (8 q 4 1 − 6 q 2 1 q 2 2 + q 4 2 ) , I = p 4 2 + m 2 q 2 2 3  4 q 1 q 2 p 1 p 2 + q 2 2 p 2 1 − ( q 2 2 − 6 q 2 1 ) p 2 2 + 1 12 q 2 2 ( q 2 2 − 2 q 2 1 ) 2  . (63) In this work, w e will show, that the ab o v e are the only in tegrable cases, w hen m 6 = 0. An imp ortant p oint to note is that there is a complete symmetry with resp ect to interc han ging Λ and λ . It is a consequence of the fact, th at there exists a canonica l transformation of the form p 1 → i p 1 , q 1 → − i q 1 , p 2 → p 2 , q 2 → q 2 , (64) that c han ges the Hamiltonia n into H = 1 2  p 2 1 + p 2 2  + 1 2  k ( q 2 1 + q 2 2 ) − m 2 q 2 1 q 2 2  + 1 4  Λ q 4 1 + λq 4 2  , (65) whic h is the same afte r s w appin g the i n dices. W e shall use this form of H , where the kinetic part is in the n atural form, to mak e the u se of some already existing th eorems more straigh tforward. 5.2 In tegrability of the reduced problem It is p ossible to giv e str ingen t conditions for integrabilit y of the system, by consider ing a reduced Hamiltonian. Namely , we can separate p oten tial V in to h omogeneous p arts of degree 2 17 and 4: V = V h 2 + V h 4 , V h 2 = 1 2 k  q 2 1 + q 2 2  , V h 4 = 1 4  − 2 m 2 q 2 1 q 2 2 + Λ q 2 1 + λq 4 2  . (66) The follo wing fact is crucial in our considerations: if a p oten tial V is in tegrable then its h ighest order as w ell as the lo w est order parts are also integrable. This fact needs some add itional justification as its sev eral kno w n pro ofs are not correct. In fact, consider p oten tial V = V min + · · · + V max , where V min and V max are homoge n eous parts of V = V ( q ), q ∈ C n of the lo west and the highest degree, resp ectiv ely . Assu me that it admits meromorphic commuting indep enden t first integ r als F 1 , . . . , F n . If F i = R i /S i for certain holomorph ic f unctions R i and S i , then we set f i = r i /s i , where r i and s i are the lo we st order terms of expansions of R i and S i in to the p o wer ser ies. It is easy to show that f i are first int egrals of V min . Ho w ever, w e cannot claim that they are functionally indep en den t. F ortunately w e can use in the d escrib ed situation the Ziglin Lemma [65] which guaran tees that w e can alw a ys c ho ose first integ rals F i in suc h a wa y that their ladin g terms f i are functionally indep enden t. A more complicated situation arises with the inte grability of V max . Here we hav e to assum e that V is in tegrable with rational first in tegrals in order to distinguish their h ighest order terms. Then w e need also an appr opriate v ersion of the Ziglin Lemma. Pro ofs of these fact s will b e published elsewhere. In our case if V giv en by (66) is in tegrable then V h 2 and V h 4 m ust also b e in tegrable. V h 2 is the p otent ial of the t w o-dimensional harmonic oscill ator, th us , it i s trivially in tegrable. Ho we ver, the h omogeneous part V h 4 giv es str ong integrabilit y restrictions for the whole p oten tial V. W e will call V h 4 the reduced p otent ial and denote it b y b V . Th u s w e effectiv ely set k = 0, and are no w in p osition to exerci se known theo rems concernin g homogeneous p oten tials dep ending on tw o v ariables. In particular the complete analysis for degree 4 h as b een completed in [45]. In order to iden tify ou r p oten tial with some of th e list giv en in that pap er, w e ha v e to c hec k ho w man y Darb oux p oint s th ere exist, and what are the v alues of parameters Λ, λ and m that giv e p oten tials equ iv alen t to particular families. W e say that a non-zero p oin t ( q 1 , q 2 ) = d is a Darb oux p oin t of th e p oten tial b V ( q 1 , q 2 ) when it satisfies the equatio n b V ′ ( d ) = γ d , (67) where γ ∈ C ∗ = C \ { 0 } . Such a p oin t corresp onds to a particular solution of th e form q ( η ) = f ( η ) d , p ( η ) = ˙ f ( η ) d , (68) with f ( η ) satisfying a differen tial equation that for a p oten tial of degree 4 tak es the form ¨ f ( η ) = − γ f ( η ) 3 . (69) As explained in Section 3, particular solutions allo w for studying the v ariational equ ations along them, and yield n ecessary conditions for existence of additional first integrals. How ever, the ma jor simp lification disco v ered in [45 ] is that add itionally there is only a fin ite n umb er of parameters’ sets (or non-equiv alen t p oten tials) corresp on ding to integ rable cases. 18 F ollo wing the cited pap er’s exp osition and n otation, w e tak e I 4 , 2 and I 4 , 3 to b e the sets of in tegrable h omogeneous p oten tials of d egree 4 w ith 2 and 3 simp le Darb oux p oin ts resp ectiv ely . W e recall also four c h aracteristic p otentia ls thereof V 3 = 1 4 aq 4 1 + 1 3 bq 3 1 q 2 + 1 4 ( q 2 1 + q 2 2 ) 2 , V 4 = 1 4 aq 4 1 + q 4 2 , V 5 = 4 q 4 1 + 3 q 2 1 q 2 2 + 1 4 q 4 2 , V 6 = 2 q 4 1 + 3 2 q 2 1 q 2 2 + 1 4 q 4 2 , (70) where a and b den ote (for the sake of this paragraph) arbitrary complex n u m b ers. W e find that our p oten tial has: 1. F our simple Darb oux p oin ts, wh en Λ( m 2 + Λ)( m 2 + λ ) 6 = 0, and Λ λ 6 = m 4 . The only in tegrable cases are: (a) λ = Λ = − 1 3 m 2 ( b V is equiv alen t to V 4 ), (b) λ = − 8 3 m 2 , Λ = − 1 6 m 2 ( b V is equ iv alen t to V 5 ), (c) λ = − 8 3 m 2 , Λ = − 1 3 m 2 ( b V is equ iv alen t to V 6 ). 2. Th ree simple Darb oux p oin ts, wh en Λ = 0, and λ ( m 2 + λ ) 6 = 0. There are n o integ r able families here as I 4 , 3 = ∅ . 3. Two s imple Darb oux p oin ts, w hen either Λ = m 4 λ and λ ( m 2 + λ ) 6 = 0, or Λ = λ = 0. Again, no integrable families are presen t here b ecause I 4 , 2 = ∅ . 4. A triple Darb oux p oint, when Λ = − m 2 . Additionally there is a simple Darb oux p oint when λ 6 = 0. The p oten tial is equiv alen t to V 3 and is only in tegrable when λ = − m 2 . There are t wo immediate implications that follo w. Firstly , that the main system itself w ith k = 0 is only in tegrable in th ose four cases, and the resp ectiv e fi rst integ rals are kno wn, as given in the table. Secondly , as it w as shown in [33] those cases are the only ones which could b e in tegrable when k 6 = 0. This happ ens b ecause the in tegrabilit y of the full p oten tial implies the in tegrabilit y of the homogeneous parts of the m aximal and minimal degree (the latter is trivially solv able in our case). As the table sho ws, when th e p otent ial is equiv alen t to V 3 (or, to b e p recise, its in tegrable sub case) or V 4 , the second first integral is kno w n; b ut V 5 and V 6 only h a v e kno wn first integrals with zero cur v ature. And as was sho wn in [12], for k = 1, the v alues of Λ and λ are those of V 5 or V 6 forbid integ rability . This is easily extended to the k = − 1 case, since after the c hange of v ariables q j → e iπ / 4 q j , p j → e − iπ / 4 p j , j = 1 , 2 , (71) w e obtain a system with the sign of k c hanged, but the ratios m 2 / Λ and m 2 /λ the same. Thus, concerning the conjecture of the qu oted p ap er, our results f or k 6 = 0 enable us to state, that it is true, w hen the rational in tegrabilit y is considered. 19 Ho w eve r , the ab ov e considerations assu me that the energy v alue is generic, so that the particular solution is a non-degenerate elliptic function. As s tressed b efore, this do es not preclude the existence of an add itional first in tegral on the ph ysically crucial zero-energy lev el. 5.3 In tegrability on the zero-energy lev el W e choose not to inv estigate the Darb oux p oin ts, but the v ariational equations d irectly , as they are considerably s impler in this case. Th e Hamilt onian equations of (20) are ˙ q 1 = p 1 , ˙ p 1 = − k q 1 + m 2 q 1 q 2 2 − Λ q 3 1 , ˙ q 2 = p 2 , ˙ p 2 = − k q 2 + m 2 q 2 1 q 2 − λq 3 2 , (72) and they admit thr ee in v ariant planes as was sho wn in [44]. T hey are Π k = { ( q 1 , q 2 , p 1 , p 2 ) ∈ C 4 | q k = 0 ∧ p k = 0 } , k = 1 , 2 , Π 3 = { ( q 1 , q 2 , p 1 , p 2 ) ∈ C 4 | q 2 = αq 1 ∧ p 2 = − αp 1 } , α 2 = m 2 + Λ m 2 + λ . (73) Ob viously t wo particular solutions are { q 1 = p 1 = 0 , q 2 = q 2 ( η ) , p 2 = q ′ 2 ( η ) } , 0 = 1 2  p 2 2 + k q 2 2 + λ 2 q 4 2  , { q 2 = p 2 = 0 , q 1 = q 1 ( η ) , p 1 = q ′ 1 ( η ) } , 0 = 1 2  + p 2 1 + k q 2 1 + Λ 2 q 4 1  , (74) and in ord er to find th e third particular solution w e mak e a canonical c hange of v ariables ( q 1 , q 2 , p 1 , p 2 ) T = B ( Q 1 , Q 2 , P 1 , P 2 ) T , (75) where symplectic matrix B has th e block s tructure B =  A O O A T  , A =  − b − a − a b  , O =  0 0 0 0  , (76) and a and b are defined by a = s m 2 + Λ 2 m 2 + λ + Λ , b = s m 2 + λ 2 m 2 + λ + Λ . (77) Let us intro duce fiv e quantit ies α 1 = 2 m 2 + λ + Λ , α 2 = 3 λ Λ + 2 m 2 ( λ + Λ) + m 4 , α 3 = p ( λ + m 2 )(Λ + m 2 ) , α 4 = λ 2 + Λ 2 − λ Λ − m 4 , α 5 = λ Λ − m 4 . (78) Then, in th e new v ariables, Hamiltonian (20) has the form H = 1 2  P 2 1 + P 2 2 + k ( Q 2 1 + Q 2 2 )  + 1 4 α 1  α 5 Q 4 1 + 2 α 2 Q 2 1 Q 2 2 + 4(Λ − λ ) α 3 Q 1 Q 3 2 + α 4 Q 4 2  , (79 ) 20 and the Hamilt onian equations read ˙ Q 1 = P 1 , ˙ P 1 = − k Q 1 − 1 α 1  α 5 Q 3 1 + α 2 Q 1 Q 2 2 + (Λ − λ ) α 3 Q 3 2  , ˙ Q 2 = P 2 , ˙ P 2 = − k Q 2 − 1 α 1  α 2 Q 2 1 Q 2 + 3(Λ − λ ) α 3 Q 1 Q 2 2 + α 4 Q 3 2  . (80) Th u s, the thir d particular solution can b e seen to b e { Q 2 = P 2 = 0 , Q 1 = Q 1 ( η ) , P 1 = Q ′ 1 ( η ) } , 0 = 1 2  P 2 1 + k Q 2 1 + α 5 2 α 1 Q 4 1  . (81) Of cours e, this is on ly v alid for α 1 6 = 0. W e inv estigate wh at h app ens w hen λ + Λ = − 2 m 2 at the end of this section. Normal v ariational equatio n s (NVE’s) along th ose three solutions (in the p osition v ariables) are ξ ′′ ( η ) =  − k + m 2 q ( η ) 2  ξ ( η ) , ξ ′′ ( η ) =  − k + m 2 q ( η ) 2  ξ ( η ) , ξ ′′ ( η ) =  − k − α 2 α 1 q ( η ) 2  ξ ( η ) , (82) where q ( η ) is one of { q 1 ( η ) , q 2 ( η ) , Q 1 ( η ) } , d ep ending on the resp ectiv e particular solution. W e will consid er the k = 0 case first. Ch anging the indep enden t v ariable to z = q ( η ) 2 , all the NVE’s are redu ced to the follo wing 2 z 2 ξ ′′ ( z ) + 3 z ξ ′ ( z ) − λ i ξ ( z ) = 0 , (83) whose solution is ξ ( z ) = z − (1 ± √ 1+8 λ i ) / 4 , (84) where w e ha v e in tro d uced three imp ortan t quantit ies λ 1 = − m 2 Λ , λ 2 = − m 2 λ , λ 3 = α 2 α 5 = 3 − 2( λ 1 + λ 2 ) + λ 1 λ 2 1 − λ 1 λ 2 . (85) Note, that if an y of Λ, λ or α 5 is zero, the corresp onding particular solution is constan t and cannot b e used to restrict the problem’s integ r abilit y . Th us, we are left with the E = k = 0 case as susp ected to b e in tegrable. When we assume k 6 = 0, or more precisely k 2 = 1, and introdu ce the same indep end en t v ariable z as b efore, the NVE’s read 2 z 2 (Λ z + 2 k ) ξ ′′ ( z ) + z (3Λ z + 4 k ) ξ ′ ( z ) + ( m 2 z − k ) ξ ( z ) = 0 , 2 z 2 ( λz + 2 k ) ξ ′′ ( z ) + z (3 λz + 4 k ) ξ ′ ( z ) + ( m 2 z − k ) ξ ( z ) = 0 , 2 z 2  α 5 α 1 z + 2 k  ξ ′′ ( z ) + z  3 α 5 α 1 z + 4 k  ξ ′ ( z ) −  α 2 α 1 z + k  ξ ( z ) = 0 . (86) First, let us observ e that unlike in the previous ca se, when any of Λ, λ or α 5 is zero, the system is not in tegrable. This happ ens, b ecause then the NVE’s b ecomes the Bessel equatio n s 2 ξ ′′ ( s ) + sξ ′ ( s ) + ( s 2 − n 2 ) ξ ( s ) = 0 , (87) 21 with n = 1 and in a new v ariable s = m p z /k (for the first t wo equatio n s) or s = m p − 2 z /k (for the thir d equation). The Bessel equation is kno wn not to p ossess Liouvillian solutions f or n = 1 [40]. T ogether with the results of th e previous section this leads u s to the follo wing lemma. Lemma 1 System (20) c onsider e d on the zer o or gene ric ener gy level with k 2 = 1 is not inte- gr able when Λ or λ is zer o. A dditional ly for λ + Λ 6 = − 2 m 2 , it is not inte gr able when λ Λ = m 4 . Assuming that none o f those constan ts is zero, w e rescale the v ariable z in t h e three equations with z → − 2 k Λ z , z → − 2 k λ z , z → 2 k α 1 α 5 z , (88) resp ectiv ely , so that all three are transformed in to a Riemann P equ ation of the form ξ ′′ ( z ) +  1 − δ − δ ′ z + 1 − γ − γ ′ z − 1  ξ ′ ( z ) +  δ δ ′ z 2 + γ γ ′ ( z − 1) 2 + β β ′ − δ δ ′ − γ γ ′ z ( z − 1)  ξ ( z ) = 0 , (89) with the follo wing pairs of exp onen ts ( δ, δ ′ ), ( γ , γ ′ ), ( β , β ′ ) at its singular p oin ts  1 2 , − 1 2  ,  1 2 , 0  ,  1 4 + 1 4 p 1 + 8 λ i , 1 4 − 1 4 p 1 + 8 λ i  , i = 1 , 2 , 3 . (90) Using Kim u ra’s resu lts on solv abilit y of the Riemann P equ ation [38] w e chec k when the differen ce of the exp onen ts giv e us cases with the necessary conditions for in tegrabilit y satisfied, and find that the parameters m ust b elong to the follo wing families λ i = l i ( l i + 1) 2 , l i ∈ Z , i = 1 , 2 , 3 . (91) These p olynomials in l i are inv arian t with resp ect to the change l → − l − 1, so it is enough to consider non-negativ e v alues only . F ur thermore, λ 1 and λ 2 cannot b e equal to zero, as m 2 6 = 0, so l 1 and l 2 need to b e strictly p ositiv e. This r esult can b e refin ed still. First, let us n otice, that λ i are not ind ep endent and the relation b et we en th em is 1 λ 1 − 1 + 1 λ 2 − 1 + 2 λ 3 − 1 = − 1 , (92) pro vided α 1 6 = 0 and α 5 6 = 0. In the ab o ve form we had to exclude the p ossibilit y of λ i = 1, so w e consider it separately . Both of λ 1 and λ 2 cannot b e equal to 1, as that would mean α 5 = 0 and w e ha ve sho wn that then the equations are non-in tegrable if add itionally α 1 6 = 0. T he α 1 = 0 case is describ ed b elo w. If only one of λ i , say λ 1 is 1, then necessarily λ 3 = 1, which follo ws from the definition (85), and the only p ossibly integ r able cases are those with λ 2 satisfying (92 ) with l 2 ≥ 2. The same holds wh en λ 1 and λ 2 are interc hanged. Also, λ 3 = 1 requir es that one of th e remaining λ i is 1. When l 1 and l 2 are tak en to b e grater than 1, λ 1 and λ 2 are p ositiv e, so th e relation (92) requires that 2 / ( λ 3 − 1) is n egativ e. This only happ ens for l 3 = 0 and it f ollo w s that l 1 = l 2 = 2, whic h is exactly the first kn o wn in tegrable case. Since 1 / ( λ 1 − 1) and 1 / ( λ 2 − 1) are p ositiv e 22 and tend to zero mon otonicall y as l i ≥ 2 tends to infinity , there are no other solutions, and no other in tegrable sets of parameters. Finally , w e turn to see what happ ens wh en α 1 = 0, i.e. Λ + λ = − 2 m 2 . T his is equiv alent to 1 λ 1 + 1 λ 2 = 2 , (93) pro vided λ 6 = 0 and Λ 6 = 0 and the same t w o conditions of (91) hold b ecause the firs t t wo v ariational equations can still b e used . It is straigh tforward to c h ec k that the only in teger solution of 1 l 1 ( l 1 + 1) + 1 l 2 ( l 2 + 1) = 1 (94) is l 1 = l 2 = 1 (so, in ciden tally , α 5 = 0), whic h w e recognise as the second case of our ta b le. 6 Conclusions The main results of our pap er can b e summarised as f ollo w s. F or the minimally coupled scala r fields, giv en by the Hamiltonian H = 1 2  − p 2 1 + 1 q 2 1 p 2 2  − k q 2 1 + Λ q 4 1 + m 2 q 2 2 q 4 1 , (95) w e ha ve : Theorem 4 F or Λ = 0 , if the system is inte gr able then ne c essarily E = k = 0 . Theorem 5 When Λ 6 = 0 , if the system is inte gr able on a generic ener gy level then e ither 1. 9 − 4 m 2 / Λ = l 2 for some l ∈ Z , or 2. k = 0 and 9 − 4 m 2 / Λ = (2 n + 1) 2 for n + 1 2 ∈ 1 3 Z ∪ 1 4 Z ∪ 1 5 Z \ Z . Conjecture 5.1 Supp ose Λ 6 = 0 , and let n b e an inte ger satisfying 9 − 4 m 2 / Λ = (2 n + 1) 2 . If the system is i nte gr able on a generic ener gy level E 6 = 0 , then either 1. n = 1 or n = − 2 ( m = 0 in b oth c ases), or 2. k = 0 and 9 − 4 m 2 / Λ = (2 l ) 2 with l an o dd inte ger, or 3. k = 0 and n + 1 2 ∈ 1 3 Z ∪ 1 4 Z ∪ 1 5 Z \ Z . Note that this is more restrictiv e than Th eorem 5, as case 1 of th is theorem admits more v alues of n th an the conjecture’s cases 1 and 2 put together. Theorem 6 F or the zer o ener gy level, and pr ovide d that Λ 6 = 0 , if the system is inte gr able then either 1. k = 0 , or 23 2. 9 − 4 m 2 / Λ = (2 n + 1) 2 , n ∈ Z . While for the conformally coupled scalar fields, giv en b y the Hamiltonian H = 1 2  − p 2 1 + p 2 2  + 1 2  k ( − q 2 1 + q 2 2 ) + m 2 q 2 1 q 2 2  + 1 4  Λ q 4 1 + λq 4 2  , (96) w e ha ve : Theorem 7 The system r estricte d to a generic ener gy level E 6 = 0 i s inte gr able if, and only if, 1. k = 0 , and its p ar ameters b elong to the four f amilies liste d i n T able 1 . Otherwise ther e exists no additional, mer omorphic inte gr al. 2. k 2 = 1 , and its p ar ameters b elong to the first two families of T able 1. Other than that, ther e exists no additional, r ational first inte gr al. The second p art of the ab o ve theorem can b e stren gthened to meromorp hic first integ rals, although not for all v alues of the parameters, as describ ed in [12]. Theorem 8 If the system r estricte d to the zer o ener g y lev el is inte gr able, then either 1. k = 0 , or 2. k 2 = 1 and its p ar ameters b elong to the first two families of T able 1, or 3. k 2 = 1 and one of { λ 1 , λ 2 } is e qual to 1, and the other satisfies the c ondition (91) with l i ≥ 2 . Otherwise, the system is not mer omorph ic al ly inte gr able. In p articular this me ans, that for k 2 = 1 if at le ast one of Λ and λ is zer o, then the system is non-inte gr able. These are, how ever, only n ecessary and not sufficient conditions, so that the system m igh t still pro ve not to b e inte grable at all. In particular, the n um erical searc h for chao s su ggests b oth the lack of global first in tegrals, and cru cial differences in the b ehavio u r of th e sy stem f or real and imaginary v alues of the v ariables. Th is migh t b e a clue, that the system m igh t ha v e fir st in tegrals wh ic h are not analytic, and thus n ot p rolongable to the complex domain. A sys tem with similar prop ert y w as stud ies b y the authors in [46]. The immed iate physical consequences of th e non-inte grability is the non-existence of con- stan ts or motion (by definition) or, in other wo r ds, la w s of conserv ation. Th is resu lts not only in th e complexit y of evolutio n but also in a harder descriptiv e approac h to a ph ysical sy stem whic h do es n ot p ossess an y global, we ll d efined, p reserv ed quan tities like total charge or sp in (in general – we ha ve not considered such quan tities in the present w ork). It is also obvio u s th at direct inte gration, or obtaining the solutions in closed forms b y means of elemen tary f unctions is out of qu estion with non-in tegrable problems. Of course, dep end ing on the p rop erties of the first in tegrals, w e might get quite different re- sults, an d the r equiremen t of meromorp hicit y or rationalit y is still very restrictiv e. As describ ed in the introd uction, this lea ve s op en the question of existence of real-analytic first integrals. 24 Also w e recall that p h ysically the scale factor a cannot ev en assume negativ e v alues, and some authors argue that when cosmolog ical (instead of conform al) time is used, the ev olution is not, i n essence, c haotic [18]. Thus, we w ould lik e to str ess that Liouvillian in tegrabilit y is a mathemat- ical p rop erty of the system, and often th e metho ds used to study it requ ire the complexifica tion of v ariables. This means that wh en restricted to the narr o w er, ph ysical domain, the dynamics migh t b e muc h sim pler. And in particular w e migh t b e interested in a particular tra jectory whose b ehavio u r is far from generic. It is no surpr ise then, th at the dynamics of ou r system when r estricted to a > 0 m igh t app ear regular. It should still b e noted that the n otion of c h aos, although fr equen tly asso ciated with the int egrabilit y , h as not ye t b een su ccessfully conflated with it. An d that a regular evol u tion is not necessarily in tegrable. Ac kno wledgemen ts A v ery sp ecial thank s to th e referees of this pap er for man y suggestions of the impr o v ements of this article. F or the fi rst three authors this r esearc h was su pp orted by grant No. N N202 2126 33 of Ministry of Science and Higher Edu cation of P oland. F or the second author th is researc h has b een partially s upp orted b y th e Europ ean Comm unity pro ject GIFT (NEST-Adven tu re Pro ject no. 5006), b y Pro jet de l’Agence National de la Rec herche “Int ´ egrabilit ´ e r ´ eelle et complexe en m ´ ecanique hamiltonienne” N ◦ JC05 − 41465 and b y the grant UMK 414-A. And f or the fourth author, by the Marie Cur ie Actions T ransfer of Knowledge pr o ject COCOS (con tract MTKD- CT-2004-5 17186). App endix A. Massless minimal field When m = 0, the Hamilton-Jacobi equation f or the main Hamiltonian (10) will b ecome separable, b ecause it can b e written as E a 2 = 1 2  ∂ W ∂ φ  2 − 1 2 a 2  ∂ W ∂ a  2 − k a 4 + Λ a 6 + ω 2 φ 2 , (97) with th e full generating fun ction S = W − E η . Assum ing W = A ( a ) + F ( φ ), equation (97) can b e solv ed with F ( φ ) = Z s 2  J − ω 2 φ 2  d φ, A ( a ) = Z s 2  Λ a 4 − k a 2 − E + J a 2  d a, (98) where J is a constan t of in tegration. T he first equ ation of motion can then b e deduced f rom ∂ W ∂ E − η = Z d a q 2  Λ a 4 − k a 2 − E + J a 2  = const , (99) 25 whic h can b e rewritten as  d a d η  2 = 2  Λ a 4 − k a 2 − E + J a 2  . (100) Or, in tro ducing a new v ariable v = a 2 , as  d v d η  2 = 8(Λ v 3 − k v 2 − E v + J ) , (101) so that th e general solution is a 2 = v = 1 2Λ ℘ ( η − η 0 ; g 2 , g 3 ) + k 3Λ , (102) where g 2 = 16 3 k 2 + 16Λ E g 3 = 32 3 Λ k E + 64 27 k 3 − 32Λ 2 J, (103) and η 0 is the constant of int egration. Of cours e, for Λ = 0, equation (101) admits solutions in terms of circular f unctions. The equation for φ ( η ) is the follo wing ∂ W ∂ J = Z d φ r 2  J − ω 2 φ 2  + Z d a a 2 q 2  Λ a 4 − k a 2 − E + J a 2  = const , (104) whic h we simplify by u sing the j ust obtained solution for v ( η ) to get const = p J φ 2 − ω 2 √ 2 J + Z d η 2 v . (105) As v is an elliptic function of order tw o, the second in tegral can b e ev aluated by means of the W eierstrass zeta and sigma functions to y ield const = p J φ 2 − ω 2 √ 2 J + 1 4 √ 2 J  ζ ( η 1 ) − ζ ( η 2 )  η + 1 4 √ 2 J ln  σ ( η − η 1 ) σ ( η − η 2 )  , (106) where η 1 , 2 , are the zeroes of v ( η ), given b y 3 ℘ ( η 1 , 2 ; g 2 , g 3 ) = − 2 k , (107) and the constant of in tegration can b e determined from the b oundary conditions on the field φ . The functions ζ and σ are defined as follo ws − ζ ′ ( z ) = ℘ ( z ) , lim z → 0  ζ ( z ) − 1 z  = 0 , σ ′ ( z ) σ ( z ) = ζ ( z ) , lim z → 0 σ ( z ) z = 1 . (108) Again, for J = 0, the integ rals in (104) reduce to simpler functions. 26 App endix B. Massless conformal field F or m = 0 w e can separately solv e equations for eac h v ariable, so that w e ha ve E 1 = − 1 2 ˙ a 2 − 1 2 k a 2 + 1 4 Λ a 4 , E 2 = 1 2 ˙ φ 2 + 1 2 ω 2 φ 2 + 1 2 k φ 2 + 1 4 λφ 4 , (109) with E 1 + E 2 = E b eing th e total energy . Th e first of these is immediately solv ed, wh en we substitute v 1 = a 2 to get ˙ v 2 1 = 2Λ v 3 1 − 4 k v 2 1 − 8 E 1 v 1 , (110) whose solution is v 1 ( η ) = 2 Λ ℘ ( η − η 1 ; g 2 , g 3 ) + 2 k 3Λ , (111) with η 1 the in tegration constan t and g 2 = 4 3 k 2 + 4Λ E 1 , g 3 = 8 27 k 3 + 4 3 k Λ E 1 . (112) Of course, when Λ = 0 the W eierstrass function ℘ reduces to a trigonometric fu nction. Similarly , for the other v ariable, we su bstitute v 2 = φ 2 and obtain ˙ v 2 2 = − 2 λv 3 − 4 k v 2 + 8 E 2 v − 4 ω 2 , (113) whose solution is v 2 ( η ) = − 2 λ ℘ ( η − η 2 ; g 4 , g 5 ) − 2 k 3 λ , (114) where g 4 = 4 3 k 2 + 4 λE 2 , g 5 = 8 27 k 3 + 4 3 k λE 2 + λ 2 ω 2 , (115) and η 2 is the in tegration constan t. As b efore, for λ = 0 the solution deg enerates to trigonometric functions. App endix C. Lam´ e equation in the Lam ´ e-Hermite case Let us consid er Lam ´ e equation d 2 y d t 2 = ( n ( n + 1) ℘ ( t ) + B ) y , n ∈ N , (116) where the W eierstrass function has tw o p erio ds 2 ω 1 and 2 ω 2 whic h are indep en den t o ver R . W e denote its differen tial Galois group o v er C ( ℘, ˙ ℘ ) b y G . F unction v = ℘ ( t ) satisfies the differenti al equation ˙ v 2 = 4 v 3 − g 2 v − g 3 =: f ( v ) (117) 27 The alg ebr aic form of Lam ´ e equation is obtain from (11 6 ) b y setting z = ℘ ( t ) and it r eads y ′′ + 1 2 f ′ ( z ) f ( z ) y ′ − n ( n + 1) z + B f ( z ) y = 0 , n ∈ N . (118) Let G AL b e the differen tial Galois group o v er C ( z ) of this equation. As it w as sho w in [20, Section 5], G = G AL ∩ SL(2 , C ), and moreo v er it w as also sho wn that G is finite iff G AL is finite. In [9, Corollary 3.4] it w as prov ed that if G AL is finite, then G AL is a dihedral group D m of order 2 m , f or a certa in m ≥ 3. In this case , G = D m ∩ SL(2 , C ) =  exp 2 π i l m 0 0 exp − 2 π i l m  | l = 0 , . . . , m − 1  . This fact implies that if G is fi nite, then it is a cyclic group of order m for a certain m ≥ 3, so there are t wo indep end en t solutions y 1 and y 2 of (116) such that y m i ∈ C ( ℘, ˙ ℘ ), for i = 1 , 2. No w, it is kno wn that for giv en n ∈ N , and m ≥ 3 the num b er of lin early non-equiv alen t Lam ´ e equations (118) with d ifferen tial Galois D m is fin ite, see [9, 24]. Nev erth eless, for long time it w as un clear if there exists a Lam ´ e equation (118) for w hic h G AL is finite. Th is p roblem, among other things, w as analysed b y Baldassarri and Dwork, s ee [2 , 3, 4], but only a b oun d on m w as foun d. Later, see [5 , 6], examples of Lam ´ e equ ations (118) with a fi nite differen tial Galois group w ere found. In pr actice, it is imp ortant to distinguish ing parameters n , B , g 2 and g 3 for w hic h G AL is D m with p rescrib ed m ≥ 3. 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