Revisiting Lie integrability by quadratures from a geometric perspective Jos´e F. Carin˜ena∗ Departamento de F´ısica Te´orica and IUMA Facultad de Ciencias, Universidad de Zaragoza Fernando Falceto† Departamento de F´ısica Te´orica and BIFI Facultad de Ciencias, Universidad de Zaragoza 7 1 Janusz Grabowski‡ 0 2 Institute of Mathematics, Polish Academy of Sciences n a Manuel F. Ran˜ada§ J Departamento de F´ısica Te´orica and IUMA 4 1 Facultad de Ciencias, Universidad de Zaragoza ] h p - h Abstract t a AfterashortreviewoftheclassicalLietheorem,afinitedimensionalLiealgebraofvector m fieldsisconsideredandthemostgeneralconditionsunderwhichthe integralcurvesofoneof [ the fields canbe obtainedby quadraturesina prescribedwaywill be discussed,determining 1 alsothenumberofquadraturesneededtointegratethesystem. Thetheorywillbeillustrated v with examples andanextensionofthe theoremwhere the Lie algebrasare replacedby some 7 distributions will also be presented. 0 9 3 1 Introduction: the meaning of Integrability 0 . 1 Integrability is a topic that has been receiving quite a lot of attention because such not clearly 0 7 defined notion appears in many branches of science, and in particular in physics. The exact 1 meaning of integrability is only well defined in each specific field and each one of the many pos- : v sibilities of definingin aprecise way the concept of integrability has a theoretic interest. Loosely i X speakingintegrability referstothepossibility of findingthesolutions of agiven differential equa- r tion (or a system of differential equations), but one may also look for solutions of certain types, a for instance, polynomial or rational ones, or expressible in terms of elementary functions. The existenceofadditionalgeometricstructuresallowsustointroduceotherconceptsofintegrability, andsothenotion of integrability isoften identifiedas complete integrability or Liouville integra- bility [A], but we can also consider generalised Liouville integrability or even non-Hamiltonian integrability [MF]. For a recent description of other related integrability approaches see e.g. [O, MCSL]. Once a definition of integrability is accepted, systems are classified into integrable and non- integrable systems. Groups of equivalence transformations allow to do a finer classification, all systemsinthesameorbitshavingthesameintegrability properties. Thereforeifsomeintegrable ∗email: [email protected] †email: [email protected] ‡email: [email protected] §email: [email protected] 1 cases have been previously selected we will have a related family of integrable cases. So, even if the generic Riccati equation is not integrable by quadratures, all Riccati equations related to inhomogeneous linear differential equations are integrable by quadratures too, and this provides us integrability conditions for Riccati equations [CL1, CLR, CR]. Theknowledge of particular solutions can also beusefulfor transformingthe original system in simpler ones, and the prototypes of this situation are the so called Lie systems admitting a superposition rule for expressing their general solutions in terms of a generic set of a finite number of solutions [CGM1, CGM2, CGM3, CGR, CIMM, CL2, CR]. This is a report of a recentcollaboration Prof. Grabowskiwithmembersof theDepartment ofTheoretical Physicsof ZaragozaUniversity[CFGR]onadifferentconceptofintegrability, themostclassicalLieconcept of integrability by quadratures, i.e. all of its solutions can be found by algebraic operations (includinginversionoffunctions)andcomputationofintegralsoffunctionsofonevariable(called quadratures). Ourapproachdoesnotresorttotheexistenceofadditionalcompatiblestructures,butsimply usesmoderntoolsofalgebraandgeometry. Inordertoavoiddependenceofaparticularchoiceof coordinates we should consider the problem from a geometric perspective, replacing the systems of differential equations by vector fields, a global concept, in such a way that the integral curves of suchvector fieldsarethesolutions of asystemof differential equations inacoordinatesystem. The two main tools to be used are finite-dimensional Lie algebras of vector fields, in particular solvable Lie algebras (see e.g. [AKN]) or nilpotent Lie algebras [MK, G], and distributions spanned by vector fields. The aim is to extend Lie classical results of integrability [AKN]. The paper is organised as follows: the fundamental notions on Lie integrabilty and their relations with the standard Arnold-Liouville integrability are recalled in Section 2 and some concepts of cohomology needed to analyse the existence of solutions for a system of first order differential equations are recalled in Section 3. The approach to integrability recently proposed in [CFGR] is sketched in section 4 and some interesting algebraic properties are studied in Section 5. Theapproach is illustrated in section 6with theanalysis, withoutany recourse to the symplectic structure, of a recent example of a Holt-related potential that is not separable but is superintegrable with high order first integrals, while the last sections are devoted to extending the previous results to the more general situation in which, instead of having a Lie algebra, L, of vector fields, we have a vector space V such that its elements donot close a finite dimensional real Lie algebra, but rather generate a general integrable distribution of vector fields. 2 Integrability by quadratures Given an autonomous system of first-order differential equations, x˙i = fi(x1,...,xN) , i = 1,...,N, (1) we can consider changes of coordinates and then the system (1) becomes a new one. This suggest that (1) can be geometrically interpreted in terms of a vector field Γ in a N-dimensional manifold M whose local expression in the given coordinates is ∂ Γ = fi(x1,...,xN) . ∂xi The integral curves of Γ are the solutions of the given system, and then integrate the sys- tem means to determine the general solution of the system. More specifically, integrability by quadratures means that you can determine the solutions (i.e. the flow of Γ) by means of a finite number of algebraic operations and quadratures of some functions. There are two main techniques in the process of solving the system: • Determination of constants of motion: Constants of motion provide us foliations such that Γ is tangent to theleaves of the foliation, and reducing in this way the problem to a family of lower dimensional problems, one on each leaf. 2 • Search for symmetries of the vector field: The knowledge of infinitesimal one-parameter groups of symmetries of the vector field (i.e. of the system of differential equations), suggests us to use adapted local coordinates, the system decoupling then into lower di- mensional subsystems. Morespecifically,theknowledgeofrfunctionallyindependent(i.e. suchthatdF ∧···∧dF 6= 1 r 0) constants of motion, F ,...,F , allows us to reduce the problem to that of a family of vector 1 r fields Γ defined in the N −r dimensional submanifolds M given by the level sets of the vector c c function of rank r, (F ,...,F ) :M → Rr. Of course the best situation is when r = N −1: the 1 r leaveseare one-dimensional, giving us the solutions to the problem, up to a reparametrisation. There is another way of reducing the problem. Given an infinitesimal symmetry (i.e. a vector field X such that [X,Γ] = 0), then, according to the Straightening out Theorem [AM80, AM88, CP], in a neighbourhoodof a pointwhereX is differentfrom zero we can choose adapted coordinates, (y1,...,yN), for which X is written as ∂ X = . ∂yN Then, the symmetry condition [X,Γ] = 0 implies that Γ has the form ∂ ∂ ∂ Γ = f¯1(y1,...,yN−1) +...+f¯N−1(y1,...,yN−1) +f¯N(y1,...,yN−1) , ∂y1 ∂yN−1 ∂yN and its integral curves are obtained by solving the system of differential equations dyi = f¯i(y1,...,yN−1) , i = 1,...,N −1 dt  dyN  = f¯N(y1,...,yN−1).  dt  We have reduced theproblem to a subsystem involving only the first N −1 equations, and once this has been solved, the last equation is used to obtain the function yN(t) by means of one more quadrature. Note that the new coordinates, y1,...,yN−1, are such that Xy1 = ··· = XyN−1 = 0, i.e. they are constants of the motion for X and therefore we cannot easily find such coordinates in a general case. Moreover, the information provided by two different symmetry vector fields cannot be used simultaneouslyinthegeneralcase,becauseitisnotpossibletofindlocalcoordinates(y1,...,yN) such that ∂ ∂ X = , X = , 1 ∂yN−1 2 ∂yN unless that [X ,X ]= 0. 1 2 In terms of adapted coordinates for the dynamical vector field Γ, i.e. Γ = ∂/∂yN, the integration is immediate, the solution curves being given by yk(t) = yk, k = 1,...,N −1, yN(t)= yN(0)+t. 0 This proves that the concept of integrability by quadratures depends on the choice of initial coordinates, because in these adapted coordinates the system is easily solved. However, it will be proved that when Γ is part of a family of vector fields satisfying appro- priate conditions, then it is integrable by quadratures for any choice of initial coordinates. Both, constants of motion and infinitesimal symmetries, can be used simultaneously if some compatibility conditions are satisfied. We can say that a system admitting r < N − 1 func- tionally independent constants of motion, F ,...,F , is integrable when we know furthermore s 1 r commuting infinitesimal symmetries X ,...,X , with r+s= N such that 1 s [X ,X ] = 0, a,b = 1,...,s, and X F = 0, ∀a = 1,...,s,α = 1,...r. a b a α 3 Theconstants ofmotion determineas-dimensionalfoliation (withs = N−r)andtheformer condition means that the restriction of the s vector fields X to the leaves are tangent to such a leaves. Sometimes we have additional geometric structures that are compatible with the dynamics. For instance, a symplectic structure ω on a 2n-dimensional manifold M. Such a 2-form relates, by contraction, in a one-to-one way, vector fields and 1-forms. Vector fields X associated with F exact 1-forms dF are said to be Hamiltonian vector fields. We say that ω is compatible means that the dynamical vector field itself is a Hamiltonian vector field X . H Particularly interesting is the Arnold–Liouville definition of (Abelian) complete integrability (r = s = n, with N = 2n) [A, AKN, VVK1, L]. The vector fields are X = X and, for a Fa instance, F = H. 1 The regular Poisson bracket defined by ω (i.e. {F ,F } = X F ), allows us to express the 1 2 F2 1 above tangency conditions as X F = {F ,F } = 0 – i.e. the n functions are constants of Fb a a b motion in involution and their corresponding Hamiltonian vector fields commute. Our aim is to study integrability in absence of additional compatible structures, the main tool being properties of Lie algebras of vector fields containing the given vector field, very much in the approach started by Lie. The problem of integrability by quadratures depends on the determination by quadratures of the necessary first-integrals and on findingadapted coordinates, or, in other words, in finding a sufficient number of invariant tensors. The set X (M) of strict infinitesimal symmetries of Γ ∈ X(M) is a linear space: Γ X (M) = {X ∈ X(M) | [X,Γ] = 0} . Γ The flow of vector fields X ∈ X (M) preserve the set of integral curves of Γ. Γ The set of vector fields generating flows preserving the set of integral curves of Γ up to a reparametrisation is a real linear space containing X (M) and will be denoted Γ XΓ(M) = {X ∈ X(M) | [X,Γ] = f Γ} , f ∈ C∞(M). X X The flows of vector fields in XΓ(M) preserve the one-dimensional distribution generated by Γ. Moreover, for any function f ∈ C∞(M), XΓ(M) ⊂ XfΓ(M), i.e. XΓ(M) only depends of the distribution generated by Γ and not on Γ itself. One can check that XΓ(M) is a real Lie algebra and X (M) is a Lie subalgebra of XΓ(M). Γ However X (M) is not an ideal in XΓ(M). Γ As indicated above, finding constants of motion for Γ is not an easy task, at least in absence ofacompatiblesymplecticstructure. However, theexplicitknowledgeoffirstintegrals ofagiven dynamicalsystem hasproved tobeof greatimportanceinthestudyofthequalitative properties of the system. The important point is that an appropriate set of infinitesimal symmetries of Γ can also provide constants of motion. More specifically, let {X ,...,X } be a set of d vector 1 d fields taking linearly independent values in every point and which are infinitesimal symmetries of Γ. If they generate an involutive distribution, i.e. there exist functions f k such that ij [X ,X ]= f kX , then, for each triple of numbers i,j,k the functions f k are constants of the i j ij k ij motion, i.e. Γ(f k) = 0. In fact, Jacobi identity for the vector fields Γ,X ,X , i.e. ij i j [[Γ,X ],X ]+[[X ,X ],Γ]+[[X ,Γ],X ]= 0, i j i j j i leads to [[X ,X ],Γ] = 0 =⇒ [f kX ,Γ]= −Γ(f k)X = 0. i j ij k ij k Moreover, for any other index l, X (f k) is also a constant of motion, because as X is a l ij l symmetry of Γ, then L L (f k) = L L (f k) =0. Γ Xl ij Xl Γ ij Theconstants ofmotionsoobtainedarenotfunctionallyindependentbutatleastthisproves (cid:0) (cid:1) (cid:0) (cid:1) theusefulnessoffindingthesefamiliesofvector fieldswhenlookingforconstants ofmotion. This points outthe convenience of extending thetheory from Lie algebras of symmetries to involutive distributions, as we will do in the final part of the paper. 4 3 Lie theorem of integrability by quadratures The first important result is due to Lie who established the following theorem: Theorem 3.1. If n vector fields, X ,...,X , which are linearly independent in each point of 1 n an open set U ⊂ Rn, generate a solvable Lie algebra and are such that [X ,X ] = λ X with 1 i i 1 λ ∈ R, then the differential equation x˙ = X (x) is solvable by quadratures in U. i 1 We only prove the simplest case n= 2. The differential equation can be integrated if we are able to find a first integral F (i.e. X F = 0), such that dF 6= 0 in U. The straightening out 1 theorem [AM80, AM88, CP], says that such a function F locally exists. F implicitly defines one variable, for instance x , in terms of the other one by F(x ,φ(x )) = k. 2 1 1 IfX andX aresuchthat [X ,X ]= λ X , andα is a1-form, defineduptomultiplication 1 2 1 2 2 1 0 by a function, such that i(X )α = 0, as X is linearly independent of X at each point, 1 0 2 1 i(X )α 6= 0, and we can see that the 1-form α = (i(X )α )−1α is such that i(X )α = 0 and 2 0 2 0 0 1 satisfies, by construction, the condition i(X )α = 1. Such 1-form α is closed, because X and 2 1 X generate X(R2) and 2 dα(X ,X )= X α(X )−X α(X )+α([X ,X ]) = α([X ,X ]) =λ α(X )= 0. 1 2 1 2 2 1 1 2 1 2 2 1 Therefore, there exists, at least locally, a function F such that α= dF, and it is given by F(x ,x ) = α, 1 2 Zγ where γ is any curve with end in the point (x ,x ). This is the function we were looking for, 1 2 because dF = α and then i(X )α = 0 ⇐⇒ X F = 0, i(X )α = 1⇐⇒ X F = 1. 1 1 2 2 Wedonotpresentheretheproofforgeneralnbecauseitappearsasaparticularcaseofthemore general situation we consider later on. The result of this theorem has been slightly generalized for adjoint-split solvable Lie algebras in [VVK2]. 4 Recalling some basic concepts of cohomology Let be g a Lie algebra and a a g-module, or in other words, a is a linear space that is carrier space for a linear representation Ψ of g, i.e. Ψ: g → Enda satisfies Ψ(a)Ψ(b)−Ψ(b)Ψ(a) = Ψ([a,b]), ∀a,b ∈ g. By a k-cochain we mean a k-linear alternating map α : g×···×g → a. If Ck(g,a) denotes the linear space of k-cochains, for each k ∈ N we define δ : Ck(g,a) → Ck+1(g,a) by (see e.g. k [CE] and [CI] and references therein) k+1 (δ α)(a ,...,a ) = (−1)i+1Ψ(a )α(a ,...,a ,...,a )+ k 1 k+1 i 1 i k+1 i=1 X + (−1)i+jα([a ,a ],a ,...,a ,...,a ,...,a ), i j 1 b i j k+1 i<j X b b where a denotes, as usual, that the element a ∈ g is omitted. i i Thelinearmapsδ canbeshowntosatisfyδ ◦δ = 0,andconsequentlythelinearoperator k k+1 k δ on Cb(g,a) = ∞k=0Ck(g,a) whose restriction to each Ck(g,a) is δk, satisfies δ2 = 0. We will then denote L Bk(g,a) = {α ∈Ck(g,a) | ∃β ∈ Ck−1(g,a) such that α= δβ} = Imageδ , k−1 Zk(g,a) = {α ∈Ck(g,a) | δα = 0} = kerδ . k 5 The elements of Zk(g,a) are called k-cocycles, and those of Bk(g,a) are called k-coboundaries. As δ is such that δ2 = 0, we see that Bk(g,a) ⊂ Zk(g,a). The k-th cohomology group Hk(g,a) is Zk(g,a) Hk(g,a) := , Bk(g,a) and we will define B0(g,a) = 0, by convention. We are interested in the case where g is a finite-dimensional Lie subalgebra of X(M), a = p(M), and consider the action of g on a given by Ψ(X)ζ = L ζ. The case p = 0, has been X used, for instance, in the study of weakly invariant differential equations as shown in [COW]. V The cases p = 1,2, are also interesting in mechanics [CI]. Coming back to the particular case p = 0, a = 0(M) = C∞(M), g = X(M), the elements of Z1(g, 0(M)) are linear maps h :g → C∞(M) satisfying V V (δ h)(X,Y)= L h(Y)−L h(X)−h([X,Y])= 0 , X,Y ∈ X(M), 1 X Y and those of B1(g,C∞(M)) are linear maps h: g → C∞(M) for which ∃g ∈ C∞(M) with h(X) = L g . X Lemma Let {X ,...,X } be a set of n vector fields whose values are linearly independent at 1 n each point of an n-dimensional manifold M. Then: 1) The necessary and sufficient condition for the system of equations for f ∈ C∞(M) X f = h , h ∈C∞(M), i= 1,...,n, i i i to have a solution is that the 1-form α ∈ 1(M) such that α(X ) = h be an exact 1-form. i i 2) If the previous n vector fields generate a n-dimensional real Lie algebra g (i.e. there exist V real numbers c k such that [X ,X ] = c kX ), then the necessary condition for the system of ij i j ij k equations to have a solution is that the R-linear function h: g → C∞(M) defined by h(X ) = h i i is a 1-cochain that is a 1-cocycle. Proof.- 1) For any pair of indices i,j, if X f = h and X f = h , then, as ∃f k ∈ C∞(M) i i j j ij such that [X ,X ] =f kX , i j ij k X (X f)−X (X f)=[X ,X ]f = f kX f =⇒ X (h )−X (h )−f kh = 0, i j j i i j ij k i j j i ij k and as α(X )= h , we obtain that as i i dα(X ,X )= X α(X )−X α(X )−α([X ,X ]) = X (h )−X (h )−f kh , i j i j j i i j i j j i ij k the 1-form α is closed. Consequently, a necessary condition for the existence of the solution of the system is that α be closed. 2) Consider a = C∞(M), g the n-dimensional real Lie algebra generated by the vector fields X ,andthecochaindeterminedbythelinearmaph: g → C∞(M). Nowthenecessarycondition i for the existence of the solution is written as: X (h )−X (h )−c kh = (δ h)(X ,X )= 0. i j j i ij k 1 i j This is just the 1-cocycle condition. Most properties of differential equations are of a local character: closed forms are locally exact and we can restrict ourselves to appropriate open subsets U of M, i.e. open submanifolds, wheretheclosed 1-formisexact, . Thenifαisclosed, itislocally exact, α= df inacertain open U, f ∈ C∞(U), and the solution of the system can be found by one quadrature: the solution function f is given by the quadrature f(x)= α, Zγx 6 where γ is any path joining some reference point x ∈ U with x ∈ U. x 0 We also remark that α is exact, α = df, if and only if α(X ) = df(X ) = X f = h , i.e. h is i i i i a coboundary, h= δf. In the particular case of the appearing functions h being constant the condition for the i existence of local solution reduces to α([X,Y])= 0, for each pair of elements, X and Y in g, i.e. α vanishes on the derived Lie algebra g′ = [g,g]. In particular when g is Abelian there is not any condition. 5 A generalisation of Lie theory of integration Consider a family of N vector fields, X ,...,X , defined on a N-dimensional manifold M and 1 N assume that they close a Lie algebra L over the real numbers [X ,X ]= c kX , i,j,k = 1,...,N, i j ij k and that, in addition, they span a basis of T M at every point x ∈ M. We pick up an element x in the family, X , the dynamical vector field. To emphasize its special roˆle we will often denote 1 it by Γ ≡ X . 1 Our goal, is to obtain the integral curves Φ : M → M of Γ t d (Γf)(Φ (x)) = f(Φ (x)), ∀f ∈ C∞(x), x ∈ M, t t dt by using quadratures (operations of integration, elimination and partial differentiation). The number of quadratures is given by the number of integrals of known functions depending on a finite number of parameters, that are performed. Γ plays a distinguished roˆle since it represents the dynamics to be integrated. Our approach is concerned with the construction of a sequence of nested Lie subalgebras L of the Lie algebra L, and it will be essential that Γ belongs to all these subalgebras. This Γ,k construction, forwhich moredetails can befoundin [CFGR], willbecarried outinseveral steps. The first one will be to reduce, by one quadrature, the original problem to a similar one but with a Lie subalgebra L of the Lie algebra L (with Γ ∈ L ) whose elements span at every Γ,1 Γ,1 point the tangent space of the leaves of a certain foliation. If iterating the procedure we end up with an Abelian Lie algebra we can, with one more quadrature, obtain the flow of the dynamical vector field. We determine thefoliation through a family of functions that are constant on the leaves. We first consider the ideal in L L = hΓi+[L,L], dimL = n , Γ,1 Γ,1 1 that, in order to make the notation simpler, we will assume to be generated by the first n 1 vector fields of the family (i.e. L = hΓ,X ,...,X i). This can always be achieved by Γ,1 2 n1 choosing appropriately the basis of L. Now take ζ in the annihilator of L , i.e. ζ is in the set L0 made up by the elements of 1 Γ,1 1 Γ,1 L∗ killing vectors of L , and define the 1-form α on M by its action on the vector fields in L: Γ,1 ζ1 α (X) = ζ (X), for X ∈ L. ζ1 1 As α (X) is a constant function on M, for any vector field in L, we have ζ1 dα (X,Y) = α ([X,Y]) = ζ ([X,Y]) = 0, for X,Y ∈ L,ζ ∈ L0 . ζ1 ζ1 1 1 Γ,1 Therefore the 1-form α is closed and by application of the result of the lemma the system of ζ1 partial differential equations X Q = α (X ), i = 1,...,n, Q ∈ C∞(M), i ζ1 ζ1 i ζ1 7 has a unique (up to the addition of a constant) local solution which can be obtained by one quadrature. Moreover, if we fix the same reference point x for any ζ , α depends linearly on 0 1 ζ1 ζ and, if γ is independent of ζ , we have that the correspondence 1 x 1 L0 ∋ ζ 7→ Q ∈ C∞(M) Γ,1 1 ζ1 defines an injective linear map. The system expresses that the vector fields in L (including Γ) are tangent to Γ,1 N[Y1] = {x |Q (x) = ζ (Y ), ζ ∈ L0 }⊂ M 1 ζ1 1 1 1 Γ,1 forany[Y ]∈ L/L . Locally, foranopenneigbourhoodU, theN[Y1]’sdefineasmoothfoliation 1 Γ,1 1 of n -dimensional leaves. 1 Now, we repeat the previous procedure by taking L as the Lie algebra and any leaf N[Y1] Γ,1 1 as the manifold. The new subalgebra L ⊂ L is defined by Γ,2 Γ,1 L = hΓi+[L ,L ], dimL = n , Γ,2 Γ,1 Γ,1 Γ,2 2 and taking ζ ∈ L0 ⊂ L∗ (the annihilator of L ), we arrive at a new system of partial 2 Γ,2 Γ,1 Γ,2 differential equations X Q[Y1] = ζ (X ), i= 1,...,n , Q[Y1] ∈ C∞(N[Y1]), i ζ2 2 i 1 ζ2 1 that can be solved with one quadrature and such Q[Y1] depends linearly on ζ . ζ2 2 It will be useful to extend Q[Y1] to U. We first introduce the map ζ2 x U ∋ x 7→ [Y ]∈ L /L 1 Γ,0 Γ,1 x x where x and [Y ] are related by the equation Q (x) = ζ (Y ), that correctly determines the 1 ζ1 1 1 x x map. Now, we define Q ∈ C∞(U) by Q (x) = Q[Y1 ](x). Note that by construction x ∈ N[Y1 ] ζ2 ζ2 ζ2 1 and, therefore the definition makes sense. The resulting function Q (x) is smooth provided the ζ2 reference point of the lemma changes smoothly from leave to leave. The construction is then iterated by defining N[Y1][Y2] = {x |Q (x) = ζ (Y ), Q (x) =ζ (Y ), with ζ ∈ L0 ,ζ ∈L0 } ⊂ M, 2 ζ1 1 1 ζ2 2 2 1 Γ,1 2 Γ,2 for [Y ] ∈ L /L and [Y ] ∈ L /L . Note that L generates at every point the tangent 1 Γ,0 Γ,1 2 Γ,1 Γ,2 Γ,2 space of N[Y1][Y2], therefore we can proceed as before. 2 The algorithm ends if after some steps, say k, the Lie algebra L = hX ,...,X i, whose Γ,k 1 nk vector fields are tangent to the n -dimensional leaf N[Y1],...,[Yk], is Abelian. In this moment the k k system of equations X Q[Y1],...,[Yk] = ζ (X ), i = 1,...,n , Q[Y1],...,[Yk] ∈ C∞(N[Y1],...,[Yk]), i ζk k i k−1 ζk k can be solved locally by one more quadrature for any ζ ∈ L∗ . k Γ,k Remarkthat, astheLiealgebraL isAbelian,theintegrability conditionisalways satisfied Γ,k and we can take ζ in the whole of L∗ instead of L0 . Then, as before, we extend the solutions k Γ,k Γ,k to U and call them Q . ζk With all these ingredients we can find the flow of Γ by performing only algebraic operations. In fact, consider the formal direct sum Ξ = L0 ⊕L0 ⊕···⊕L0 ⊕L∗ , Γ,1 Γ,2 Γ,k Γ,k that, as one can check, has dimension n. 8 The linear maps L0 ∋ ζ 7→ Q ∈ C∞(U) can be extended to Ξ so that to any ξ ∈ Ξ we Γ,i i ζi assign a Q ∈ C∞(U). Now consider a basis ξ {ξ ,...,ξ } ⊂ Ξ. 1 n The associated functions Q ,j = 1,...,n are independent and satisfy ξj ΓQ (x) = ξ (Γ), j = 1,2,...,n, ξj j where it should be noticed that as Γ ∈ L for any l = 0,...,k, the right hand side is well Γ,l defined, and we see from here that in the coordinates given by the Q ’s the vector field Γ has ξj constant components and, then, it is trivially integrated Q (Φ (x)) = Q (x)+ξ (Γ)t. ξj t ξj j Now, with algebraic operations, one can derive the flow Φ (x). Altogether we have performed t k+1 quadratures. 6 Algebraic properties The previous procedure works if it reaches an end point (i.e. if there is a smallest non negative integer k > 0 such that L = hΓi+[L ,L ], Γ,k Γ,k−1 Γ,k−1 is an Abelian algebra). In that case we say that (M,L,Γ) is Lie integrable of order k+1. The content of the previous section can, thus, be summarized in the following Proposition 6.1. If (M,L,Γ) is Lie integrable of order r, then the integral curves of Γ can be obtained by r quadratures. We will discuss below some necessary and sufficient conditions for the Lie integrability. Proposition 6.2. If (M,L,Γ) is Lie integrable then L is solvable. Proof.- Let L be the elements of the derived series, L = [L ,L ], L = L, (note (i) (i+1) (i) (i) (0) that L = L ). Then, (i) 0,i L ⊂ L , (i) Γ,i and if the system is Lie integrable (i.e. L is Abelian for some k), then we have L = 0 Γ,k (k+1) and, therefore, L is solvable. Proposition 6.3. If L is solvable and A is an Abelian ideal of L, then (M,L,Γ) is Lie integrable for any Γ ∈A. Proof.- Using that A is an ideal containing Γ, we can show that A+L = A+L . Γ,i (i) We proceed again by induction: if the previous holds, then A+L = A+[L ,L ]= A+[A+L ,A+L ] = Γ,i+1 Γ,i Γ,i Γ,i Γ,i = A+[A+L ,A+L ]= A+L . (i) (i) (i+1) Now L is solvable if some L = 0 and therefore L ⊂ A, i.e. it is Abelian and henceforth the (k) Γ,k system is Lie integrable. Note that the particular case A= hΓi corresponds to the standard Lie theorem. Nilpotent algebras of vector fields also play an interesting role in the integrability of vector fields. Proposition 6.4. If L is nilpotent, (M,L,Γ) is Lie integrable for any Γ ∈ L. 9 Proof.- Let us consider the central series L(i+1) = [L,L(i)] with L(0) = L. Now, L nilpotent means that there is a k such that L(k) =0. It is easy to see, by induction, that L ⊂ hΓi+L(i) Γ,i and therefore L = hΓi is Abelian and the system is Lie integrable. Γ,k From the previous propositions, we can derive the following Corollary 6.5. Let (M,L,Γ) be Lie integrable of order r. Then: (a) If r is the minimum positive integer such that L = 0, then r ≥ r . s (rs) s (b) If L is nilpotent r is the smallest natural number such that L(rn) = 0, r ≤ r . n n 7 An interesting example We now analyse the particular case of a recently studied superintegrable system [CCR], where we dealt with an example of a potential that is not separable but is superintegrable with high order first integrals [PW], by studying limits of some potentials related to Holt potential [H]. Even if the system is Hamiltonian, that is, the dynamical vector field Γ = X is obtained from H a Hamiltonian function H by making use of a symplectic structure ω defined in a cotangent 0 bundle T∗Q we deliberately forget this fact and analyse the situation by simply considering this system just as a dynamical system (without mentioning the existence of a symplectic structure) and focusing our attention on the Lie algebra structure of the symmetries. Suppose that the dynamics is given by the vector field Γ = X defined in M = R2×R2 with 1 coordinates (x,y,p ,p ) given by x y ∂ ∂ k ∂ 2k x+k ∂ 2 2 3 Γ = X = p +p − + , 1 x y ∂x ∂y y2/3∂p 3 y5/3 ∂p x y where k and k are arbitrary constants. 2 3 Consider in this case the following three vector fields: 6x 6 ∂ ∂ X = 6p2 +3p2 +k +k +(6p p +9k y1/3) 2 x y 2y2/3 3y2/3 ∂x x y 2 ∂y (cid:18) (cid:19) 6 ∂ x 1 ∂ − k p + 4k −3 p , 2 x 2 y y2/3 ∂p y5/3 y2/3 ∂p x (cid:18) (cid:19) y 8(k x+k ) ∂ X = 4p3 +4p p2 + 2 3 p +12k y1/3p 3 x x y y2/3 x 2 y ∂x (cid:18) (cid:19) ∂ 1 ∂ + 4p2p +12k y1/3p −4k p2 x y 2 x ∂y 2y2/3 x ∂p x + (cid:0) 8k2x+k3p2 −4k (cid:1)1 p p −12k2 1 ∂ , 3 y5/3 x 2y2/3 x y 2y1/3 ∂p (cid:18) (cid:19) y and k +k x ∂ X = 6p5 +12p3p2 +24 3 2 p3 +108k y1/3p2p +324k2y2/3p 4 x x y y2/3 x 2 x y 2 x ∂x (cid:18) (cid:19) ∂ k ∂ + 6p4p +36k y1/3p3 −6 2 p4 −972k3 x y 2 x ∂y y2/3 x 2 ∂p (cid:18) (cid:19) x (cid:16) k +k x k(cid:17) 1 ∂ + 4 3 2 p4 −12 2 −108k2 p2 . y5/3 x y2/3 2y1/3 x ∂p (cid:18) (cid:19) y In order to apply the theory developed above, it suffices to compute the commutation rela- tions among the fields: [X ,X ]= 0, [X ,X ] =1944k3Γ, [X ,X ]= 432k3X (2) 2 3 2 4 2 3 4 2 2 together with: [X ,X ] = 0, i = 2,3,4. (3) 1 i 10