Natural connections for semi-Hamiltonian systems: The case of the $epsilon$-system

Natural connections for semi-Hamiltonian systems: The case of the ǫ -system P aolo Lorenzon i ∗ , Marco Pedroni ∗∗ * Dipartimen to di Matematica e Applicaz ioni Uni vers it ` a di Milano-Bicocca V ia Roberto Cozzi 53, I-201 25 Milano, Italy paolo.lo renzon i@unimib .it ** Dipartimento di Inge gneria dell’Informaz ione e Metodi M atematici Uni vers it ` a di Ber gamo V iale Marco ni 5, I-24044 Dalmine BG, Italy marco.ped roni@unib g.it Abstract Giv en a s emi-Hamiltonia n system, we constr uct an F -manifol d with a conne ction satisfy ing a suitabl e compatibility condition with the pro duct. W e ex emplify this proce- dure in th e case o f the so-ca lled ǫ -system. The corres pondin g connection turns out to be flat, and the flat coord inates gi ve rise to addition al chains of commuting flows. 1 Introduction This paper is a natural continuati on of [12] and deals with F -manifold s and their associated integrable hierarchies. The aim of [12] ess entially was to extend the concept of F -m anifold with compat ible connection to the non-flat case and to show its relev ance in t he th eory of integrable systems of hydrodynamic ty pe. T he integrability condi tion (i.e., the condit ion en- suring the existence of i ntegrable flo ws ) in such a frame work becomes the fol lowing simple requirement on the connection ∇ and on the structure constants c i j k entering t he defi nition of these F -manifol ds: R k lmi c n pk + R k lip c n mk + R k lpm c n ik = 0 , (1.1) where R i j k l is the Riemann tens or of the connection ∇ . Thus, the s tarting point in [12] was an F -m anifold, and the g oal was the associated integrable hierarchy . The approach of this paper 1 is different, since here we m ove in th e opposite direction. The starting point is an i ntegrable hierarchy of hydrodynamic type, and the goal is the construction of an F -manifold with compatible connection. More precisely , we cons ider the case of semi-Hamiltoni an systems , that is, diagonal systems of hydrodynamic type [20], u i t = v i ( u ) u i x , u = ( u 1 , . . . , u n ) , i = 1 , . . . , n, (1.2) whose coef ficients v i ( u ) (usually called c ha racteristic velocities ) satisfy the system of e qua- tions ∂ j  ∂ k v i v i − v k  = ∂ k  ∂ j v i v i − v j  ∀ i 6 = j 6 = k 6 = i, (1.3) where ∂ i = ∂ ∂ u i . Equati ons (1.3) are the integrability conditions for three d iffe rent systems: the first one, giv en by ∂ j w i w i − w j = ∂ j v i v i − v j , (1.4) provides the char acteristic velocities of the symmetries u i τ = w i ( u ) u i x i = 1 , ..., n, of (1.2); the second one is the system ∂ i ∂ j H − Γ i ij ∂ i H − Γ j j i ∂ j H = 0 , Γ i ij = ∂ j v i v j − v i , (1.5) whose solutions H are t he conserved densities of ( 1.2); the third one is ∂ j ln √ g ii = ∂ j v i v j − v i , (1.6) and relates the characteristi c ve locities of the system wi th a class of diago nal metrics. These metrics and the associated Levi-Ci vit a connection play a crucial role [3] in the Hamilt onian formalism of (1.2). The first result of thi s p aper is the existence of a connection ∇ canoni cally associated with a gi ven semi-Hami ltonian system , to be called the natural connectio n of the s ystem. By definition, it is comp atible—in the sense of (1.1)—wi th the product c i j k = δ i j δ i k , and leads t o a structure of F -manifold with comp atible connection. W e poi nt out that the natural connec- tion does not coincide , in general, with the Levi-Ci vita connections of the diagonal metrics satisfying the system (1.6)—whose Christof fel symbols are the Γ i ij appearing in (1.5). This alternative app roach to th e theory of sem i-Hamiltoni an syst ems m ight be useful not onl y from a geom etric v iewpoint, but also for applications. Indeed, it happens that the connection ∇ menti oned above t urns ou t to be flat ev en in cases where the soluti ons o f (1.6) neither are flat nor have t he Egorov property 1 . In su ch a si tuation the metrics that are 1 W e re call that a metric is said to have the Ego rov p roper ty (or to be p otential) if th ere exist coor dinates such that the metric is diagonal and g ii = ∂ i φ . In other words, the r otation coefficients β ij = ∂ i √ g j j √ g ii (1.7) must be symmetric 2 compatible with the conn ection ∇ are n ot in variant with r espect to the pr oduct . Nev ertheless the flatness of the connection and the condition ∇ l c i j k = ∇ j c i lk , entering the definition of F -manifold wi th compatib le connection, are suf ficient to defi ne t he so-called principal hierarchy , as it was shown in [12] extending a s tandard constructio n of Dubrovin. As an example, we study the semi-Hamiltonian system u i t = u i − ǫ n X k =1 u k ! u i x , i = 1 , . . . , n, (1.8) known in the literature as ǫ -system. It has been studied b y several authors. In particul ar , we refer to [5, 15 , 19] for the case ǫ = − 1 , to [6] for the case ǫ = − 1 2 , to [4] for the case ǫ = 1 , and to [18, 11, 10] for the general case. For n > 2 the metrics g ii = ϕ i ( u i ) h Q l 6 = i ( u i − u l ) 2 i ǫ , i = 1 , . . . n, (1.9) (where ϕ i ( u i ) are arbitrary non-vanishing functions of a single variable) satisfying the s ystem (1.6) are not of Egorov type. Indeed, their rotati on coef ficients β ij = " Q l 6 = i ( u i − u l ) 2 Q l 6 = j ( u j − u l ) 2 # ǫ 2 ǫ u j − u i (1.10) are not symmetric 2 . Therefore the natural connection of the ǫ -system does not coi ncide with the Le vi -Civita con nection of any of the metrics (1.9 ). In the second half of the paper we study in details such nat ural connection, that turns out to be flat. Thus, besides the usual higher flo ws of (1.8), we obtain ( n − 1) additi onal chains of commuti ng flows associated with th e non-trivial flat coordinates of the natural c onnection. Remarkably , the recursi ve identities relating these flo ws ha ve a double geometrical interpre- tation: t he first one, in t erms of the usual recursive procedure in volved in the constructi on of the principal h ierarchy; t he second one, in term s of certain cohomological equations ap- pearing in [11, 10]. Finall y , we explicitly construct t he associated principal hierarchy in the cases n = 2 and n = 3 . 2 In [10] it was observed that they satisfy the Egorov-Da rboux system ∂ k β ij = β ik β kj i 6 = j 6 = k X k ∂ k β ij = 0 i 6 = j X k u k ∂ k β ij = − β ij i 6 = j if the ϕ i are constant. 3 Ackno wledgments W e thank Andrea Raimondo for useful discussi ons; many ideas of the present paper have their orig in i n [12], writ ten in collaborati on with him. M.P . would like to thank the De- partment Matemat ica e Applicazi oni of t he Mi lano-Bicocca Un iv ersity for the hospit ality . This work has b een partially suppo rted by t he European Communit y through (ESF) Sci- entific Programme MISGAM . Some of t he comp utations have been performed wi th Maple Software. 2 F -manif o lds with com patible co nnection and r e lated in- tegrable systems In this section we revie w the most imp ortant results of [12]. First of all let u s rec all the main definition of that paper . Definition 2.1 An F -manifol d with compatible connection is a manifold endowed with an associative commut ative multi plicative structure given by a (1 , 2) -tensor field c and a t or- sionless connection ∇ s atisfying condition ∇ l c i j k = ∇ j c i lk (2.1) and condition R k lmi c n pk + R k lip c n mk + R k lpm c n ik = 0 , (2.2) wher e R k ij l = ∂ j Γ k li − ∂ l Γ k j i + Γ k j m Γ m li − Γ k lm Γ m j i is the Riemann tensor of ∇ . Definition 2.2 An F -manifol d with compatible connection is called semisim ple if ar ound any point ther e ex ist coor dinat es ( u 1 , . . . , u n ) —called canon ical coordinates —such t hat th e structur es constants become c i j k = δ i j δ i k . On the loop space o f a sem isimple F -mani fold wi th compatible connection, one can define a semi-Hamilton ian hi erarchy . The flows of t his hierarchy are u i t = c i j k X k u j x , (2.3) where the vector fi elds X satis fy the system c i j m ∇ k X m = c i k m ∇ j X m . (2.4) Using the generalized hod ograph method [20], one obtains the general solutio n of (2.3) in implicit form as tX i + xe i = Y i , (2.5) 4 where e = n X i =1 ∂ ∂ u i is t he unit y of th e algebra and Y is an arbit rary solution of (2.4). Notice that the above representation of the solu tion in terms of critical po ints of a fa mily of vector fields—as well as the expressions (2.3,2.4)—holds true in an y coordinate system. If the connection ∇ is flat, the definition of F -manifold with compati ble connectio n reduces to the following definit ion, due to Manin [14]. Definition 2.3 An F -mani fold wi th compatible flat connection is a manifo ld endowed with an as sociative commutat ive multipli cative st ructur e given by a (1 , 2) -tenso r field c and a torsionless flat connection ∇ sat isfying condition (2.1). In flat coordinates, condition (2.1) reads ∂ l c i j k = ∂ j c i lk . This, together with the commutativity of the algebra, im plies that c i j k = ∂ j C i k = ∂ j ∂ k C i . Therefore, condit ion (2.1 ) is equiv alent to the local existence of a vector field C satisfying, for any pair ( X, Y ) of flat vector fields, the condition [14]: X ◦ Y = [ X, [ Y , C ]] , where ( X ◦ Y ) k = c k ij X i Y j . The h ierarchy associated with an F -manifold with compatible flat con nection i s usuall y called principal hierar chy . It can been defined in the following way , which is a straightfor- ward generalization of the original definitio n given b y Dubrovin in the case of Frobenius manifolds. First of all, one defines the so-called primary flows : u i t ( p, 0) = c i j k X k ( p, 0) u j x , (2.6) where ( X (1 , 0) , . . . , X ( n, 0) ) is a basis of fl at vector fields. Then, starting from these flows, one can define the “higher flo ws” of the hierarchy , u i t ( p,α ) = c i j k X k ( p,α ) u j x , (2.7) by means of the following recursive relati ons: ∇ j X i ( p,α ) = c i j k X k ( p,α − 1) . (2.8) Remark 2.4 The vec tor fields obtained b y me ans of the re cursive r ela tions (2.8 ) ar e nothing but th e z -coeffic ients of a basis of flat vector field s of t he deformed connecti on [2] d efined, for any pair of vector fields X and Y , by ˜ ∇ X Y = ∇ X Y + z X ◦ Y , z ∈ C . In the following section we will show how to const ruct a semis imple F -m anifold with compatible connection starting from a semi-Hamiltoni an sy stem. 5 3 Fr om se mi-Ham iltonian system to F -m anif old s w ith com- patible connection : The natur al connection Let u i t = v i ( u ) u i x , u = ( u 1 , . . . , u n ) , i = 1 , . . . , n, (3.1) be a semi-Hamilt onian syst em, that is, suppo se that the characteristic velocities v i ( u ) satisfy (1.3). W e want to define a semisi mple F -manifold w ith compatibl e connection whose asso- ciated semi-Hamiltonian syst em cont ains (3.1) . First of all , we define the structure const ants c i j k simply assign ing to t he Riemann in variants u i the role of canonical coordinates. This means that in the coordinates ( u 1 , . . . , u n ) we hav e c i j k = δ i j δ i k . (3.2) Once give n the structure constants, the definition of the connection ∇ is quit e rigid , apart from the freedom i n the choice o f the Christoffel s ymbols Γ i ii . Indeed, such a connection must be torsion-free, Γ i j k = Γ i k j , and must satisfy condition (2.1), that in canonical coordinates reduces to Γ i j k = 0 for i 6 = j 6 = k 6 = i, (3.3) Γ i j j = − Γ i j i for i 6 = j . (3.4) Moreover , the s pace of solutions of (2.4) must contai n the characteristic velocities of t he semi-Hamilton ian s ystem we started with. Putting X i = v i in (2.4), we obtain Γ i j i = ∂ j v i v j − v i for i 6 = j , (3.5) where the v i satisfy (1.3). The compatibi lity condition (2.2) between c and ∇ is now auto- matically satisfied, thanks to the following l emmas. Lemma 3.1 The only non-vanishing components of the Riemann tenso r of a connection sat- isfying conditions (3.3,3.4,3.5) ar e R i ik i = − R i iik = ∂ k Γ i ii − ∂ i Γ i ik (3.6) and R i q q i = − R i q iq = ∂ q Γ i q i − ∂ i Γ i q q +  Γ i iq  2 + Γ i q q Γ q q i − n X p =1 Γ i pi Γ p q q . (3.7) Pr oof . First of all we hav e R i q k l = 0 for distinct indices (3.8) R i ik l = − R i ilk = ∂ k Γ i il − ∂ l Γ i ik = 0 if i 6 = k 6 = l 6 = i (3.9) R i q k i = − R i q ik = ∂ k Γ i iq + Γ i ik Γ i iq − Γ i iq Γ q q k − Γ i ik Γ k k q = 0 if i 6 = q 6 = k 6 = i. (3.10) 6 The first identity is a con sequence of the vanishing of t he Christoffel symbol s Γ i j k when the three indices are distinct, the second one is a consequence of the s emi-Hamiltoni an prop erty (1.3), and the third one is a consequence of the identity (see [20]) ∂ k Γ i iq + Γ i ik Γ i iq − Γ i iq Γ q q k − Γ i ik Γ k k q = v k − v i v q − v k  ∂ q  ∂ k v i v k − v i  − ∂ k  ∂ q v i v q − v i  and of th e semi-Hamil tonian property . Finally , using (3.4) and (3.10) one can easily prove that R i q q l = − R i q lq = − ∂ l Γ i q q + Γ i q q Γ q q l − Γ i il Γ i q q − Γ i ll Γ l q q = 0 if i 6 = q 6 = l 6 = i. (3.11 ) All the other components vanish apart from (3.6 ,3.7).  Lemma 3.2 Conditio n (2.2) follows fr om (3.3), (3.4), and (3.5). Pr oof . In canonical coordinates, conditio n (2.2) tak es the form R n lmi δ n p + R n lip δ n m + R n lpm δ n i = 0 . (3.1 2) It is clearly satisfied i f n 6 = p, m, i . Since p, m, i appear cyclicly in (3.12), it is suffi cient to prove it for p = n , that is, R n lmi + R n lin δ n m + R n lnm δ n i = 0 . (3.13) In turn, this cond ition needs a check only for i 6 = n , l eading to R n lmi + R n lin δ n m = 0 . W e end up with R n lmi = 0 , where i 6 = n and m 6 = n , which is the content of Lemm a 3.1.  Remark 3.3 Let us consider a diagonal m etric sol ving t he system (1.6). Its Levi-Civita connection clearly sati sfies (3.3) and (3.5). It fu lfills a lso (3.4) if a nd o nly i f the metric is potential in the coor dinat es ( u 1 , . . . , u n ) . Ther efor e, only in th is case one c an choose the Γ i ii in suc h a way that ∇ is such a L evi-Civita conn ection. In other wor d s, a connection ∇ satis- fying conditions (3.3,3.4,3.5) does not necessar ily coincide with the Levi-Civita connection of a metric solving (1.6). A (counter)e xample is g iven by the ǫ -system discussed in Section 5. A natural way to elimi nate the resi dual freedom in the choice of the Christoffel coef fi- cients Γ i ii ( i = 1 , . . . , n ) is to impose the additi onal requirement ∇ e = 0 , (3.14) where e = P n i =1 ∂ ∂ u i is the unity of the algebra. Indeed, condition (3.14) means that n X k =1 Γ i j k = 0 . (3.15) If i 6 = j , it coincides with (3.4). If i 6 = j , it gives Γ i ii = − X k 6 = i Γ i ik , i = 1 , . . . , n. (3.16) 7 Definition 3.4 W e call the connection ∇ defin ed by conditions (3.3,3.4,3.5,3.16) the natural connection associated with the semi-Hamiltoni an s ystem (3.1). By constructi on, t he natural connection and the product (3.2) sati sfy Definit ion 2.1 of F - manifold with compatible connection, and the semi -Hamiltonian system (3.1) i s one of the integrable flo ws associated with this F -manifold. Remark 3.5 W e added condition (3.16) in or der to associate a un ique con nection to a given semi-Hamiltoni an system. W e will see i n Section 5 that this i s a very conv enient choice for the ǫ -system. Neve rtheless, ther e might be situati ons wher e a d iffer ent condi tion has to b e chosen. W e close this section with a remark on the Euler vector field E = n X k =1 u k ∂ ∂ u k . (3.17) Pr oposition 3.6 The Euler v ector field E and the uni ty of the algebra e satisf y the identity ∇ e E = e, wher e ∇ is the natural connection. Pr oof . W e ha ve that ( ∇ e E ) i = e k ∇ k E i = e k  ∂ k E i + Γ i k l E l  = n X k =1  δ i k + Γ i k l E l  = 1 + n X k =1 Γ i k l ! E l = 1 = e i , where we ha ve used (3.15).  4 Special recurr e nce r elations for semi-Ham iltonian sys - tems In view of th e example of the ǫ -system (to be d iscussed in the next section ), we recall some results obtained in [10, 11]. Let M be an n -dim ensional manifold. A tensor field L : T M → T M of type (1 , 1) is said to be torsionless if the identity [ LX, LY ] − L [ LX , Y ] − L [ X , LY ] + L 2 [ X , Y ] = 0 8 is verified for any pair of vector fields X and Y on M . According to the theory of graded deriv ations of Fr ¨ olicher-Nijenhuis [7], a torsionless tensor field L of type (1 , 1) defines a diffe rential operator d L on the Grassmann algebra of di f ferential forms on M , satisfying the fundamental conditions d · d L + d L · d = 0 , d L 2 = 0 . On functions and 1-forms this deri vation is defined b y the following equati ons: d L f ( X ) = d f ( LX ) (that is, d L f = L ∗ d f ) d L α ( X , Y ) = Lie LX ( α ( Y )) − Lie LY ( α ( X )) − α ([ X , Y ] L ) , where [ X , Y ] L = [ LX, Y ] + [ X , LY ] − L [ X , Y ] . For instance, if L = diag( f 1 ( u 1 ) , . . . , f n ( u n )) , the actio n of d L on functions is given by the formula d L g = n X i =1 f i ∂ g ∂ u i du i . W e assume a : M → R t o be a function which satisfies the cohomological condition dd L a = 0 , (4.1) and, from now on, that L = diag ( u 1 , . . . , u n ) . Then, according to the results of [10], to any solution h = H ( u ) of the equation dd L h = dh ∧ da (4.2) we can associate a sem i-Hamiltoni an hierarchy . Indeed, it is easy to prov e that the system of quasilinear PDEs u i t =  − ∂ i K ∂ i H  u i x , i = 1 , . . . , n, (4.3) is semi-Hamilto nian for any solut ion h = K ( u ) of the equation (4.2), and that, o nce fixed H , the flows associated to any pair ( K 1 , K 2 ) of s olutions of (4.2) commute. Moreover , there is a recursi ve procedure to obtain solutio ns of (4.2) . Lemma 4.1 ([1, 13]) Let K 0 be a solution of ( 4.2 ) . Then the functi ons K α r ecursively d e- fined by dK α +1 = d L K α − K α da, α ≥ 0 , (4.4) satisfy equation (4.2). 9 Let us illus trate ho w to apply the previous procedure in the case of the (trivial) solutions H = a and K 0 = − a o f equation (4.2). Using the recursiv e relations (4.4), we get u i t 0 = − ∂ i K 0 ∂ i a u i x = u i x u i t 1 = − ∂ i K 1 ∂ i a u i x = [ u i − a ] u i x = [ u i + K 0 ] u i x u i t 2 = − ∂ i K 2 ∂ i a u i x = [( u i ) 2 + K 0 u i + K 1 ] u i x . . . u i t n = − ∂ i K n ∂ i a u i x = [( u i ) n + K 0 ( u i ) n − 1 + K 1 ( u i ) n − 2 + · · · + K n − 1 ] u i x . . . The choice H = − K 0 = a = ǫ T r ( L ) = ǫ P n i =1 u i giv es rise to the ǫ -system discussed in the Introduction and in the following secti on. 5 The ǫ -system In this s ection we exemplify ou r construction in the case of the ǫ -syst em (1.8). In partic- ular , we show that its natural con nection is flat, so that we ob tain an F -manifold wi th flat compatible connection and its principal hierarchy . 5.1 The natural connection of the ǫ -system According to Definition 3.4, the natural connection of the ǫ -system is gi ven by Γ i j k = 0 for i 6 = j 6 = k 6 = i Γ i j i = ǫ u i − u j for i 6 = j Γ i j j = − Γ i j i = ǫ u j − u i for i 6 = j Γ i ii = − X k 6 = i Γ i ik = − X k 6 = i ǫ u i − u k . (5.1) Pr oposition 5.1 The natural connection of t he ǫ -system is flat. Pr oof . W e ha ve that R i ik i = ∂ k Γ i ii − ∂ i Γ i ik = ∂ k − X j 6 = i ǫ u i − u j ! − ∂ i  ǫ u i − u k  = − ǫ ( u i − u k ) 2 + ǫ ( u i − u k ) 2 = 0 10 and R i q q i = ∂ q Γ i q i − ∂ i Γ i q q +  Γ i iq  2 + Γ i q q Γ q q i − n X p =1 Γ i pi Γ p q q = ∂ q Γ i q i − ∂ i Γ i q q + Γ i iq (Γ i iq − Γ q iq ) − X p 6 = i,q Γ i pi Γ p q q − Γ i ii Γ i q q − Γ i q i Γ q q q = ( ∂ q + ∂ i )  ǫ u i − u q  + 2 ǫ 2 ( u i − u q ) 2 + X p 6 = i,q ǫ 2 ( u i − u p )( u p − u q ) + ǫ u i − u q X p 6 = i,q  − ǫ u i − u p + ǫ u q − u p  + ǫ u i − u q  − ǫ u i − u q + ǫ u q − u i  = X p 6 = i,q ǫ 2 ( u i − u p )( u p − u q ) + ǫ u i − u q X p 6 = i,q ǫ ( u i − u p − u q + u p ) ( u i − u p )( u q − u p ) = 0 . Due to Lemma 3.1, th ere are no more comp onents of R to be checked, so that ∇ is flat.  The next proposi tion is dev oted to the relations betw een the natural con nection (5.1) a nd the Euler vector field (3.17). Pr oposition 5.2 The co vari ant derivative of E is given by ∇ j E k = (1 − nǫ ) δ k j + ǫ , tha t is, ∇ E = (1 − nǫ ) I + ǫe ⊗ d (T rL) , (5.2) wher e I is the identity on the tangent bundle and e is the u nity . Ther efore , for any ve ctor field X , ∇ X ( E − ǫ (T rL) e ) = (1 − nǫ ) X . (5.3) Mor eover , the Euler vector field E is linear in flat coor dinat es, i.e ., ∇∇ E = 0 . (5.4) Pr oof . Formula (5.2) follows from the very definitions of ∇ and E . Then, us ing the flatness of e , we obtain (5.3). Finally , ∇∇ E = ∇ ((1 − nǫ ) I + ǫe ⊗ d ( T rL)) = 0 since also ∇ I = 0 (due to the fact that ∇ is torsionless ) and ∇ ( d (T rL)) = 0 (as noticed i n the next subsection, before Proposition 5.4).  W e remark that (5.4) i s on e of th e property entering the defi nition of Frobenius manifold. 11 5.2 Flat coordinates In this subsection we discus s som e properties of the flat coordinates of t he natural connection (5.1) of the ǫ -system. W e have to find a basis of flat exact 1-forms θ = θ i du i , that is, n independent solut ions of the linear system of PDEs ∂ j θ i − ǫ θ i − θ j u i − u j = 0 , i = 1 , . . . , n, j 6 = i ∂ i θ i + ǫ X k 6 = i θ k − θ i u k − u i = 0 , i = 1 , . . . , n, (5.5) which is equiv alent to ∂ j θ i − ǫ θ i − θ j u i − u j = 0 , i = 1 , . . . , n, j 6 = i n X k =1 ∂ k θ i = 0 , i = 1 , . . . , n. (5.6) In particular , we hav e th at 0 = n X k =1 ∂ k θ i = n X k =1 ∂ i θ k = ∂ i n X k =1 θ k ! , showing t hat P n k =1 θ k is constant if θ = θ k du k is flat. Remark 5.3 It trivia lly follows fr om (5.6) that f is a flat coor din ate if and only if ( u i − u j ) ∂ j ∂ i f − ǫ ( ∂ i f − ∂ j f ) = 0 , i = 1 , . . . , n, j 6 = i n X k =1 ∂ k ∂ i f = 0 , i = 1 , . . . , n. (5.7) Since ( dd L f − d ( ǫ T r L ) ∧ d f ) ij = ( u i − u j ) ∂ j ∂ i f − ǫ ( ∂ i f − ∂ j f ) , any flat coordinate of the n atural connection of t he ǫ -system s olves equation (4.2) with a = − ǫ T r L = − ǫ P n i =1 u i . A trivial soluti on of the sy stem (5.6 ) is give n by θ j = 1 for all j , corresponding to the flat 1-form θ (1) = P n j =1 du j = d f 1 ǫ , w here f 1 ǫ = P n j =1 u j . The oth er flat coordinates can b e chosen according to Pr oposition 5.4 Ther e exist flat coor dinat es ( f 1 ǫ , f 2 ǫ , . . . , f n ǫ ) whose partial derivatives ∂ i f p ǫ ( u ) ar e homogeneous functi ons of de gr ee − nǫ for all p = 2 , . . . , n and i = 1 , . . . , n . In partic- ular , if ǫ 6 = 1 n ther e e xist fla t coor dinates ( f 1 ǫ , f 2 ǫ , . . . , f n ǫ ) such th at f p ǫ ( u ) is a homogeneous function of de gr ee (1 − nǫ ) for all p = 2 , . . . , n . 12 Pr oof . Suppo se that φ = φ j du j is a flat 1 -form. Then P n j =1 φ j = c constant, and θ := φ − c n θ (1) is still a flat form and satisfies P n j =1 θ j = 0 . Then equations (5.5) entail that n X j =1 u j ∂ j θ i = − nǫθ i , so that θ i ( u ) is hom ogeneous o f d egree − nǫ for all i . This shows that w e can always find a basis  θ (1) , θ (2) , . . . , θ ( n )  of flat forms such that the component s of θ ( p ) , for al l p ≥ 2 , are homogeneous of degree − n ǫ . Since flat forms are exact, the first assertion is proved. The second assertio n simpl y follows from the general fact that a function f ( u ) , whose partial deri vati ves are homogeneous of de gree r 6 = − 1 , is (up to an additive constant) homo- geneous of degree ( r + 1) . E ven though this is well k nown, we give a proof for t he reader’ s sake. W e know that n X k =1 u k ∂ k ( ∂ j f ) = r ∂ j f , therefore we ha ve ∂ j n X k =1 u k ∂ k f − ( r + 1) f ! = 0 , meaning that n X k =1 u k ∂ k f = ( r + 1) f + c for some constant c that can be eliminated if r 6 = − 1 .  In the case n = 2 , we hav e that θ (2) = ( u 1 − u 2 ) − 2 ǫ ( du 1 − du 2 ) , so that the flat coordinates are f 1 ǫ = u 1 + u 2 , f 2 ǫ = ( u 1 − u 2 ) 1 − 2 ǫ if ǫ 6 = 1 2 (5.8) f 1 ǫ = u 1 + u 2 , f 2 ǫ = ln ( u 1 − u 2 ) if ǫ = 1 2 . (5.9) In the case n = 3 , assumin g ǫ 6 = 1 3 and taking i nto account t he homo geneity of t he flat coordinates, it is possi ble to reduce the system (5.7 ) to a third order ODE whos e solut ions can be explicitly written in terms o f hypergeometric functions 2 F 1 ( α ; β ; γ ; z ) (see the Appendi x 13 for more details). If ǫ 6 = 1 3 we obtain the flat coordinates (in the domain where u 3 > u 1 ) f 1 ǫ = u 1 + u 2 + u 3 f 2 ǫ = (1 − 3 ǫ )(2 u 2 − u 3 − u 1 )[( u 3 − u 1 )( u 1 − u 2 ) 2 ] − ǫ 2 F 1  ǫ ; 1 − ǫ ; 1 + 2 ǫ ; u 2 − u 3 u 1 − u 3  + (1 + ǫ )[( u 3 − u 1 )( u 1 − u 2 ) 2 ] − ǫ ( u 1 − u 3 ) 2 F 1  2 − ǫ ; ǫ − 1 ; 1 + 2 ǫ ; u 2 − u 3 u 1 − u 3  f 3 ǫ = (2 u 2 − u 1 − u 3 )[( u 3 − u 1 )( u 3 − u 2 ) 2 ] − ǫ 2 F 1  ǫ ; 1 − ǫ ; 1 − 2 ǫ ; u 3 − u 2 u 3 − u 1  + − [( u 3 − u 1 )( u 3 − u 2 ) 2 ] − ǫ ( u 3 − u 1 ) 2 F 1  ǫ − 1; 2 − ǫ ; 1 − 2 ǫ ; u 3 − u 2 u 3 − u 1  . It turns out that i n the case ǫ = 1 3 the function s f 2 ǫ and f 3 ǫ reduce to a cons tant. For i nteger values of the parameter ǫ one obtains si mpler expressions. For i nstance, in the case ǫ = − 1 (up to inessential constant fac tors) we have f 1 ǫ = u 1 + u 2 + u 3 f 2 ǫ = 4( u 3 − u 1 ) 3 ( u 1 + u 3 − 2 u 2 ) f 3 ǫ = 4( u 3 − u 2 ) 3 ( u 2 + u 3 − 2 u 1 ) , and in the case ǫ = 2 we have f 1 ǫ = u 1 + u 2 + u 3 f 2 ǫ = 4( u 2 + u 3 − 2 u 1 ) ( u 2 − u 1 ) 3 ( u 3 − u 1 ) 3 f 3 ǫ = 4( u 1 + u 2 − 2 u 3 ) ( u 3 − u 2 ) 3 ( u 3 − u 1 ) 3 . This concludes the discussi on of t he case ǫ 6 = 1 3 . W e will make later some consideration for the case ǫ = 1 3 . Remark 5.5 Gi ven any set ( f 1 ǫ , . . . , f n ǫ ) of flat coor di nates, the na tural connection ∇ is the Levi-Civita conn ection of the metr ic η = η ij d f i ǫ ⊗ d f j ǫ for any choice of the i n vertible symmetric matrix ( η ij ) . F or n = 2 , choosing f 1 ǫ and f 2 ǫ as above, and ( η ) ij =  0 1 1 0  , we obtain g ii = ( − 1) i 2(2 ǫ − 1 ) | u 1 − u 2 | 2 ǫ , i = 1 , 2 , (5.10) which i s one of the metrics (1.9). On the contrary—as we have said in the Intr oduction— the metr ics (1.9 ) ar e not of Egor ov type i f n > 2 . This means that for n > 2 the natural connection for the ǫ -system d oes not coincide with t he Levi-Civita connection of the metrics (1.9). 14 5.3 The structur e constants Let us discus s in details the case n = 2 . In the coordi nates (5.8 ) and (5.9) t he structu re constants are giv en by c 1 11 = c 2 12 = c 2 21 = 1 2 , c 2 11 = c 1 12 = c 2 12 = c 2 22 = 0 , c 1 22 = ( f 2 ǫ ) 4 ǫ 1 − 2 ǫ 2(2 ǫ − 1 ) 2 for ǫ 6 = 1 2 and by c 1 11 = c 2 12 = c 2 21 = 1 2 , c 2 11 = c 1 12 = c 2 12 = c 2 22 = 0 , c 1 22 = 1 8 f 2 ǫ for ǫ = 1 2 . H ence the vector potential C has components C 1 = ( f 2 ǫ ) − 2 2 ǫ − 1 4(2 ǫ + 1) + 1 4 ( f 1 ǫ ) 2 C 2 = 1 2 f 1 ǫ f 2 ǫ for ǫ 6 = ± 1 2 and C 1 = 1 16 f 2 ǫ  ln f 2 ǫ − 1  + 1 4 ( f 1 ǫ ) 2 C 2 = 1 2 f 1 ǫ f 2 ǫ for ǫ = ± 1 2 . It is easy to check that, lowering the index of the vector potential with the metric (5.10), that i n fla t coordinates i s ant idiagonal with components g 12 = g 21 = 1 , we obtain an exact 1- form. In other words, for n = 2 we obtain a scalar po tential F satis fying WD VV equ ations: F = 2 ǫ − 1 4(2 ǫ + 1)(2 ǫ − 3) ( f 2 ǫ ) − 2 ǫ − 3 2 ǫ − 1 + 1 4 ( f 1 ǫ ) 2 f 2 ǫ , if ǫ 6 = ± 1 2 , 3 2 (5.11) F = 1 16 ( f 2 ǫ ) 2 ln f 2 ǫ − 3 32 ( f 2 ǫ ) 2 + 1 4 ( f 1 ǫ ) 2 f 2 ǫ , if ǫ = ± 1 2 (5.12) F = 1 16 ln f 2 ǫ + 1 4 ( f 1 ǫ ) 2 f 2 ǫ , if ǫ = 3 2 (5.13) Let us finally consider the case n = 3 , ǫ = 1 . As flat coordinates we can choose f 1 = u 1 + u 2 + u 3 f 2 = 1 2 ( u 1 − u 2 ) ( u 3 − u 1 ) f 3 = 1 2 ( u 2 − u 3 ) ( u 1 − u 2 ) . 15 In such coordinates the structure constants read: c 1 11 = 1 3 , c 2 11 = 0 , c 3 11 = 0 , c 1 12 = 0 , c 2 12 = 1 3 , c 3 12 = 0 , c 1 13 = 0 , c 2 13 = 0 , c 3 13 = 1 3 c 1 22 = − 1 12 f 3 (3 ( f 2 ) 2 + 3 f 2 f 3 + ( f 3 ) 2 ) ( f 2 ) 3 ( f 2 + f 3 ) 3 c 2 22 = − 1 24 (10 ( f 2 ) 2 + 9 f 2 f 3 + 2 ( f 3 ) 2 ) ( f 3 ) 2 √ 2 ( f 2 + f 3 ) 2 f 2 q − f 3 f 2 ( f 2 + f 3 ) ( f 3 + 2 f 2 ) 2 c 3 22 = − 1 8 ( f 3 + 2 f 2 ) 2 f 3 3 √ 2 ( f 2 ) 2 ( f 2 + f 3 ) 2 q − f 3 f 2 ( f 2 + f 3 ) ( f 3 + 2 f 2 ) 2 c 1 23 = 1 12  f 2 + f 3  − 3 c 2 23 = − 1 24 ( f 2 ) 2 (4 ( f 2 ) 2 + 12 f 2 f 3 + 5 ( f 3 ) 2 ) √ 2 ( f 2 + f 3 ) 2 f 3 q − f 3 f 2 ( f 2 + f 3 ) ( f 3 + 2 f 2 ) 2 c 3 23 = 1 24 (10 ( f 2 ) 2 + 9 f 2 f 3 + 2 ( f 3 ) 2 ) ( f 3 ) 2 √ 2 ( f 2 + f 3 ) 2 f 2 q − f 3 f 2 ( f 2 + f 3 ) ( f 3 + 2 f 2 ) 2 c 1 33 = − 1 12 f 2 (( f 2 ) 2 + 3 f 2 f 3 + 3 ( f 3 ) 2 ) ( f 3 ) 3 ( f 2 + f 3 ) 3 c 2 33 = 1 8 ( f 3 + 2 f 2 ) ( f 2 + 2 f 3 ) ( f 2 ) 3 √ 2 ( f 3 ) 2 ( f 2 + f 3 ) 2 q − f 3 f 2 ( f 2 + f 3 ) ( f 3 + 2 f 2 ) 2 c 3 33 = 1 24 ( f 2 ) 2 (4 ( f 2 ) 2 + 12 f 2 f 3 + 5 ( f 3 ) 2 ) √ 2 ( f 2 + f 3 ) 2 f 3 q − f 3 f 2 ( f 2 + f 3 ) ( f 3 + 2 f 2 ) 2 W e do not di splay the components of the vector potential C , since t he c orresponding e x pres- sions are quite cumbersome. 5.4 The primary flows In order t o define t he primary flows we need a basis of flat vector fields X = X i ∂ ∂ u i , t hat is, n independent solutions of the linear system of PDEs ∂ j X i + ǫ X i − X j u i − u j = 0 , i = 1 , . . . , n, j 6 = i ∂ i X i − ǫ X k 6 = i X k − X i u k − u i = 0 , i = 1 , . . . , n (5.14) 16 which is equiv alent to ∂ j X i + ǫ X i − X j u i − u j = 0 , i = 1 , . . . , n, j 6 = i (5.15) [ e, X ] = 0 . (5.16) Comparing (5.14 ) wi th (5.5), one noti ces that the components X i of a flat vector fields for ǫ are g iv en b y the components of a flat 1-form for − ǫ . Therefore, from Proposition 5 .4 we hav e that there alw ays exists a basis of flat vector fields  X (1) = e, X (2) , . . . , X ( n )  such that the components X i ( p ) ( u ) , for p = 2 , . . . , n , are h omogeneuos functions of degree nǫ . In the case n = 2 we ha ve, for ǫ 6 = − 1 2 , d f 1 − ǫ = du 1 + d u 2 d f 2 − ǫ = (1 + 2 ǫ )( u 1 − u 2 ) 2 ǫ ( du 1 − du 2 ) and therefore X (1 , 0) = ∂ ∂ u 1 + ∂ ∂ u 2 = e X (2 , 0) = (1 + 2 ǫ )( u 1 − u 2 ) 2 ǫ  ∂ ∂ u 1 − ∂ ∂ u 2  . In canonical coordinates the primary flows are thus given by u 1 t (1 , 0) = u 1 x u 2 t (1 , 0) = u 2 x and u 1 t (2 , 0) = (1 + 2 ǫ )( u 1 − u 2 ) 2 ǫ u 1 x u 2 t (2 , 0) = − (1 + 2 ǫ )( u 1 − u 2 ) 2 ǫ u 2 x . The case n = 3 , ǫ 6 = − 1 3 can be treated similarly since we know the flat coordinates. Let us consider t he case n = 3 , ǫ = − 1 3 . One of the flat vector fields is the uni ty e of the algebra. W e know th at th ere exist two other flat vector fields X (2) and X (3) , whos e components are homogeneous functions of degree -1, u 1 ∂ 1 X k ( i ) + u 2 ∂ 2 X k ( i ) + u 3 ∂ 3 X k ( i ) = − X k ( i ) , i = 2 , 3 , k = 1 , 2 , 3 , (5.17) satisfying the additional property X 1 ( i ) + X 2 ( i ) + X 3 ( i ) = 0 , i = 2 , 3 . (5.18) Since ∂ j X k ( i ) = ∂ k X j ( i ) , from (5.17) we obtain u 1 X 1 ( i ) + u 2 X 2 ( i ) + u 3 X 3 ( i ) = c, (5.19) 17 where c is a con stant. T w o cases are possib le: c = 0 and c 6 = 0 . In both cases, taking into account condit ion (5.18), we can write one of the components of t he vector field X ( i ) in terms of t he remaining two. Substitutin g th e result in (5.14), we obtain a sys tem of 3 equations whose solution is X 1 (2) = ( u 2 − u 3 ) 1 / 3 ( u 3 − u 1 ) 2 / 3 ( u 1 − u 2 ) 2 / 3 X 2 (2) = ( u 3 − u 1 ) 1 / 3 ( u 2 − u 3 ) 2 / 3 ( u 1 − u 2 ) 2 / 3 X 3 (2) = ( u 1 − u 2 ) 1 / 3 ( u 3 − u 1 ) 2 / 3 ( u 2 − u 3 ) 2 / 3 for c = 0 and X 1 (3) = c u 2 − u 1 + c ( u 3 − u 2 ) 1 / 3 3( u 3 − u 1 ) 2 / 3 ( u 1 − u 2 ) Z du 3 ( u 3 − u 2 ) 1 / 3 ( u 3 − u 1 ) 1 / 3 X 2 (3) = c u 2 − u 1 + c ( u 3 − u 1 ) 2 / 3 3( u 3 − u 2 ) 2 / 3 ( u 1 − u 2 ) Z du 3 ( u 3 − u 1 )( u 3 − u 2 ) 1 / 3 X 3 (3) = c 3( u 3 − u 2 ) 2 / 3 ( u 3 − u 1 ) 2 / 3 Z du 3 ( u 3 − u 2 ) 1 / 3 ( u 3 − u 1 ) 1 / 3 for c 6 = 0 . Notice that we can choose the constants of integration in the above integrals in such a way that the X i (3) be homogeneous of degree -1. Hence we can explicitly construct the principal hierar chy (2.7) also in the case ǫ = − 1 3 . 5.5 The higher flows The higher flo ws are defined by vector fi elds X ( p,α ) satisfying ∇ j X i ( p,α ) = c j ik X k ( p,α − 1) (5.20) or , more explicitly , ∂ j X i ( p,α ) + ǫ X i ( p,α ) − X j ( p,α ) u i − u j = 0 , i = 1 , . . . , n, j 6 = i (5.21) ∂ i X i ( p,α ) − ǫ X k 6 = i X k ( p,α ) − X i ( p,α ) u k − u i = X i ( p,α − 1) , i = 1 , . . . , n. (5.22) T aking into account (5.21), condition (5.22) can be written as n X k =1 ∂ k X i ( p,α ) = X i ( p,α − 1) , i = 1 , . . . , n (5.23) 18 or , in compact form, as [ e, X ( p,α ) ] = X ( p,α − 1) . (5.24) W e show now t hat—apart from s ome critical values of ǫ —the higher flows of the principal hierarchy can be obtained by appl ying t he recursi ve procedure described in Section 4. First of all, we recall from Remark 5 .3 that t he flat coordinates of the natural connection of the ( − ǫ ) -system satisfy equation (4.2), wit h L = dia g ( u 1 , . . . , u n ) and a = ǫ T r L . Therefore, they can be u sed as starting point for the recursive procedure (4.4), giving ri se to the flows (4.3), with H = a . Pr oposition 5.6 Suppose that  f 1 − ǫ = T r L, f 2 − ǫ , . . . , f n − ǫ  be the flat coor dina tes descri bed in Pr opositio n 5.4 of the natural connectio n of the ( − ǫ ) -system. If K ( p,α ) ar e the fu nctions defined r ecursively by K ( p, 0) = − ǫf p − ǫ , dK ( p,α +1) = d L K ( p,α ) − ǫK ( p,α ) d (T r L ) , α ≥ 0 , (5.25) and Y i ( p,α ) = − ∂ i K ( p,α ) ∂ i a = − 1 ǫ ∂ i K ( p,α ) , α ≥ 0 , (5.26) ar e the components of the vector fields of the corr esponding hierar chy , then the vector fields X (1 ,α ) = 1 Q α j =1 ( j − n ǫ ) Y (1 ,α ) (for ǫ 6 = j n with j = 1 , . . . , α ) and X ( p,α ) = 1 α ! Y ( p,α ) , for p = 2 , . . . , n , sat isfy the r ecursion r elations (5.20). Pr oof . W e know that from Lemma 4.1 that the function K ( p,α ) satisfies equation (4.2). Then it is easily checked t hat the vector fields Y ( p,α ) and X ( p,α ) satisfy equatio n (5.21), so that there are only relations (5.23) to be prov ed. Let us consider the case p = 1 . After writing (5.25) in canonical coordinates, ∂ j K (1 ,α +1) = u j ∂ j K (1 ,α ) − ǫK (1 ,α ) , α ≥ 0 , (5.27) and recalling that K (1 , 0) ( u ) = − ǫ P n i =1 u i , it is clear that one can show by inductio n that the partial deriv ativ es ∂ j K (1 ,α ) ( u ) are hom ogeneous functions of degree α , so that K (1 ,α ) ( u ) is homogeneous of degree ( α + 1) . Using this fact, again from (5.27) we ha ve that n X j =1 ∂ j K (1 ,α +1) = n X j =1  u j ∂ j K (1 ,α ) − ǫK (1 ,α )  = ( α + 1 − nǫ ) K (1 ,α ) , so that n X j =1 ∂ j Y i (1 ,α +1) = ( α + 1 − nǫ ) Y i (1 ,α ) , i = 1 , . . . , n, (5.28) and relations (5.23) for X i (1 ,α ) follow . 19 The case p = 2 , . . . , n can be treated i n t he same way . The only difference is that t he degree o f homo geneity of ∂ j K ( p,α ) is ( α + nǫ ) , s o that K ( p,α ) is homogeneou s of degree ( α + 1 + nǫ ) if α 6 = − 1 − nǫ .  As a consequence of the above proposition we ha ve that, if ǫ 6 = k n for a ll k ∈ N , the flows u i t (1 ,α ) = X i (1 ,α ) u i x = Y i (1 ,α ) Q α j =1 ( j − nǫ ) u i x = − ∂ i K (1 ,α ) ǫ Q α j =1 ( j − nǫ ) u i x , (5.29) u i t ( p,α ) = X i ( p,α ) u i x = Y i ( p,α ) α ! u i x = − ∂ i K ( p,α ) ǫ α ! u i x , p 6 = 1 , (5.30) (with i = 1 , . . . , n and α ≥ 0 ) define the principal hierarchy of the ǫ -system. If ǫ = k n for some k ∈ N , all t he flows (5.30) and the flo ws (5.29) with α = 0 , . . . , k − 1 st ill belong to t he princip al hierarchy . Even though the l atter is well defined, relations (5.29) do not m ake sense for α ≥ k , since the d enominator vanishes. The p oint is that th e vector field Y (1 ,k ) is flat, as one can immediately check using (5.28), and its compo nents are homogeneous o f degree k = ǫn . Therefore Y (1 ,k ) is a linear combination (with constant coef ficients) o f the flat ho mogeneous vector fields X (2 , 0) , . . . , X ( n, 0) . This m eans that Y (1 ,α ) is, for α ≥ k , a linear combi nations of the vector fields Y ( p,α − k ) , with p = 2 , . . . , n . In order to obtain t he m issing flows o f t he p rincipal hi erarchy , associated to the vector fields X (1 ,α ) with α ≥ k , one h as to solve the system (5.21,5.22) with p = 1 , α ≥ k and X (1 ,k − 1) = 1 Q k − 1 j =1 ( j − k ) Y (1 ,k − 1) . For instance, in the ca se ǫ = 1 2 , n = 2 one can immedi ately check that the v ector field Y 1 (1 , 1) = u 1 − ǫ T r L = u 1 − u 2 2 Y 2 (1 , 1) = u 2 − ǫ T r L = u 2 − u 1 2 is flat, unlike the ve ctor field X 1 (1 , 1) = 1 2 ( u 1 − u 2 ) ln  u 1 − u 2  + 3 2 u 2 − 1 2 u 1 + c 1 Y 1 (1 , 1) + c 2 X 2 (1 , 1) = − 1 2 ( u 1 − u 2 ) ln  u 1 − u 2  + 3 2 u 1 − 1 2 u 2 + c 1 Y 2 (1 , 1) + c 2 ( c 1 and c 2 are arbitrary constants), obtained solvin g the system ∂ 2 X 1 (1 , 1) + 1 2 X 1 (1 , 1) − X 2 (1 , 1) u 1 − u 2 = 0 ∂ 1 X 2 (1 , 1) + 1 2 X 1 (1 , 1) − X 2 (1 , 1) u 1 − u 2 = 0 ∂ 1 X 1 (1 , 1) − 1 2 X 1 (1 , 1) − X 2 (1 , 1) u 1 − u 2 = 1 ∂ 2 X 2 (1 , 1) − 1 2 X 1 (1 , 1) − X 2 (1 , 1) u 1 − u 2 = 1 . 20 T o conclude, we observe that, under su itable assu mptions, the recursion relations (5.25) can be written in a more explicit form. Indeed, we have the fol lowing Pr oposition 5.7 The r ecursion r elati ons (5.25) ar e algebraically solved by K (1 ,α ) = 1 α + 1 n X j =1 ( u j ) 2 ∂ j K (1 ,α − 1) − ǫ X j =1 u j ! K (1 ,α − 1) ! (5.31) and, for α 6 = − 1 − nǫ , by K ( p,α ) = 1 α + 1 + nǫ n X j =1 ( u j ) 2 ∂ j K ( p,α − 1) − ǫ X j =1 u j ! K ( p,α − 1) ! , p = 2 , . . . , n. (5.32) Pr oof . It suffic es to m ultiply (5. 27) by u j and to su m over j , t aking into account the homo- geneity of the functions K ( p,α ) .  6 A ppend ix Let us consider the system ∂ i f = θ i , (6.1) ∂ j θ i − ǫ θ i − θ j u i − u j = 0 , i = 1 , 2 , 3 , j 6 = i (6.2) θ 1 + θ 2 + θ 3 = 0 , (6.3) u 1 θ 1 + u 2 θ 2 + u 3 θ 3 = (1 − 3 ǫ ) f , (6.4) providing th e homogeneou s flat coordinates f for the natu ral connection of the ǫ -sys tem in the case n = 3 , ǫ 6 = 1 3 . Using (6.3) and (6.4) we can write two of the components of θ , for instance θ 1 and θ 3 , in terms of the remaining one and of the flat coordinate f : θ 1 = ( u 3 − u 2 ) θ 2 + (1 − 3 ǫ ) f u 1 − u 3 (6.5) θ 3 = ( u 2 − u 1 ) θ 2 − (1 − 3 ǫ ) f u 1 − u 3 . (6.6) Hence, using (6.2) with i = 2 , j = 1 , we o btain f in terms of θ 2 : f = ( u 3 − u 2 )( u 3 − u 1 ) ∂ 1 θ 2 + ǫ (2 u 3 + u 1 + u 2 ) θ 2 ǫ (1 − 3 ǫ ) . (6.7) In this way equ ation (6.2) with i = 2 , j = 3 reduces to a PDE in volving onl y the unknown function θ 2 , ( u 2 − u 3 ) ∂ 3 θ 2 + ( u 2 − u 1 ) ∂ 1 θ 2 − 3 ǫθ 2 = 0 , (6.8) 21 whose solution is giv en by θ 2 = G ( u 2 , ν )( u 1 − u 2 ) − 3 ǫ (6.9) where ν = u 3 − u 2 u 1 − u 2 and G ( u 2 , ν ) is an arbitrary function. Substitut ing (6.9) i n (6.7) and the r esult in the equation ( u 1 − u 3 ) ∂ 1 ∂ 3 f + ǫ ( ∂ 1 f − ∂ 3 f ) = 0 we obtain the third order ODE G ′′′ + (4 ν +5 ǫν − 4 ǫ − 2) ν ( ν − 1) G ′′ + (9 ǫν 2 − 13 ǫ 2 ν +2 ν 2 − 2 ν − 2 ǫ +7 ǫ 2 ν 2 +4 ǫ 2 − 9 ǫν ) ν 2 ( ν − 1) 2 G ′ + 3 ǫ 2 (2 ǫ − ǫν − ν ) ν 2 ( ν − 1) 2 G = 0 , where G ′′′ , G ′′ , G ′ are the deriv atives of G ( u 2 , ν ) with respect to ν . The above equation can be explicitly solved in term s of hyper geometric functions: G = G 1 ( u 2 )( ν − 1) ǫ ν − 2 ǫ + G 2 ( u 2 )( ν − 1) − ǫ ν − 2 ǫ 2 F 1  ǫ ; 1 − ǫ ; 1 + 2 ǫ ; ν ν − 1  + G 3 ( u 2 )( ν − 1) − ǫ 2 F 1  ǫ ; 1 − ǫ ; 1 − 2 ǫ ; ν ν − 1  , where G 1 , G 2 , G 3 are arbit rary functions of a s ingle var iable. The choice G 2 = G 3 = 0 giv es rise t o f = 0 , while the choices ( G 2 =constant, G 1 = G 3 = 0 ) and ( G 3 =constant, G 1 = G 2 = 0 ) give rise to the homogeneous flat coordinates f 2 ǫ = (1 − 3 ǫ )(2 u 2 − u 3 − u 1 )[( u 3 − u 1 )( u 1 − u 2 ) 2 ] − ǫ 2 F 1  ǫ ; 1 − ǫ ; 1 + 2 ǫ ; u 2 − u 3 u 1 − u 3  + (1 + ǫ )[( u 3 − u 1 )( u 1 − u 2 ) 2 ] − ǫ ( u 1 − u 3 ) 2 F 1  2 − ǫ ; ǫ − 1 ; 1 + 2 ǫ ; u 2 − u 3 u 1 − u 3  f 3 ǫ = (2 u 2 − u 1 − u 3 )[( u 3 − u 1 )( u 3 − u 2 ) 2 ] − ǫ 2 F 1  ǫ ; 1 − ǫ ; 1 − 2 ǫ ; u 3 − u 2 u 3 − u 1  + − [( u 3 − u 1 )( u 3 − u 2 ) 2 ] − ǫ ( u 3 − u 1 ) 2 F 1  ǫ − 1; 2 − ǫ ; 1 − 2 ǫ ; u 3 − u 2 u 3 − u 1  . as one can check by a st raightforward computation, sub stitutin g in the equations (6. 1,6.2,6.3,6.4). References [1] A. Di makis, F . M ¨ uller-Hoissen, Bi-differ ential calculi and inte grable models , J . Phys. A: Math. Gen. 33 (2000), 957–974. 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