Leonard Euler: addition theorems and superintegrable systems

Leonard Euler: addition theorems and sup erin tegrable sy stems A V Tsigano v St.Petersbur g State University, St.Petersbur g, R ussia e–mail: tsiganov@mph.phys.spbu.ru Abstract W e consider the Euler approach to construction and to in vestigation of th e sup erin tegrable systems related to th e addition theorems. As an example w e reconstruct Drac h systems and get some new tw o-dimensional sup erin tegrable St ¨ ac kel systems. P ACS n umbers: 02.30.Jr, 02.30.Ik, 03.65.Fd Mathematics Sub ject Classification: 70H06, 70H20, 35Q72 1 In tro duction There are ma n y differ e nt mathematical a pp r oac hes to c la ssification and to investigation of the super- int eg rable sy stems with very extensive literature on the sub ject, s ee s ome recent rewiews [5, 12, 15]. Here w e wan t to discuss one of the oldest but almost completely forgotten approaches, which rela t es sup e rin tegrable systems with the addition theorems. The stor y beg an in 176 1 when Euler pro posed a g eneric construction of the additional algebra ic int eg ral of equations of motion κ 1 dx 1 p P ( x 1 ) = κ 2 dx 2 p P ( x 2 ) , where P is an arbitr a ry quartic and κ ’s are integer [8]. In fact, the L a place integral for the K epler mo del and en tries of the angula r momen tum fo r the oscilla tor are particular cases of th e Euler integral. Remind, that with geometric point o f view Euler (and later Abel in his famous theorem) studied the p oin ts of intersection of the hyper e llipt ic curve µ 2 = P ( x ) with any ar bitrary alge br aical curve whose equation in a r ational form may b e written as F ( x 1 , . . . , x n ) = 0 (see the following symbolic picture for the Eule r tw o-dimensio nal case) [3 ]. In the Jacobi separa tion o f v aria bles method thes e p oin ts are the po in ts of intersection of the sep a- r ate d r elations p 2 j = P ( x j ) de fined on the whole phase spa ce with the clas sical tr aje ct o ry of motion F ( x 1 , . . . , x n ) = 0 o n its configura tional subspace. So, E uler pro posed a lg ebraic cons tr uction of the classical tr a jectories of motion without any inte- gration and in version o f the Ab el map. This particular a lgebraic construction o f the tra jectories of motion was one of the starting p oin ts of the Ja cobi inv estigations o f the la st multiplier theor y and of the inv ersion of the Abe l map. 1 Such as algebra ic curve F ( x 1 , . . . , x n ) = 0 has many differen t parameter iz ations x k ( t ), ther e are many differen t s uperintegrable systems as sociated with this curve. F or instance, the overwhelming ma- jority of the k no wn tw o-dimensional super in tegrable s y stems are related with the differen t par ametriza- tions of the conic sections . Remark 1 F or the ellipse F ( x 1 , x 2 ) = x 2 1 a 2 + x 2 2 b 2 − 1 = 0 , we can use well known para metrizations by trigonometr ic or elliptic functions x 1 = a sin t, x 2 = b cos t or x 1 = a sn e t, x 2 = b cn e t which may be asso ciated with the os cillator and the Kepler pro ble m, res pectively . 1.1 Additional in tegr a ls of motion and angle v ariables A classical sup erin tegrable system is an integrable n -dimens io nal Ha milt o nia n system with n + k sing le v alue d and functiona lly indep enden t integrals of motion on the whole phase space M . At k = n − 1 we hav e maximally sup erin tegr able systems with the clo s ed tra jector ies of motion. The Lio uville clas sical theorem on completely integrable Hamiltonian systems implies that a lm o st all p oints o f the manifold M a r e covered by a system of o pen tor oidal do m a ins with the actio n-angle co ordinates I = ( I 1 , . . . , I k ) and ω = ( ω 1 , . . . , ω n ): { I j , I k } = { ω i , ω k } = 0 , { I i , ω j } = δ ij . (1.1) The n integrals of motio n H 1 , . . . , H n are functions of the actio n v a riables I 1 , . . . , I n such that the corres p onding Ja c obian doe s no t equal to zero det J 6 = 0 , where J ij = ∂ H i ( I 1 , . . . , I n ) ∂ I j . (1.2) The n − 1 a dditional integrals of motio n φ j = X k  J − 1  kj ω k , { H 1 , φ j } = 0 , j = 2 , . . . , n, (1.3) are functiona lly indep enden t on H 1 ( I ) , . . . , H n ( I ) a nd any completely integrable system is maxima lly sup e rin tegrable system in a neighbor hoo d of every reg ular po in t of M [24]. In generic c ase the action v ar iables I k are single-v alued functions, whereas the angle v aria bles ω k are m ulti-v alued functions o n M . How ever, additional functionally independent in tegrals of motion K i hav e to be so me functions on φ j , which are sums of the multiv alued functions ω k . So, for any sup erin tegra ble system we hav e the following ques tio n: Ho w are si ngle-v alued functions obtain ed from the sums of multi-v alued func- tions? In fact for all the known s uperintegrable systems we hav e only one answer: W e hav e to apply addition theorem and to get one multi-v alued function on the singl e-v alued argumen t, w hic h is the desi r ed in tegral of moti on: X k  J − 1  kj ω k = φ i ( K j ) , ⇒ { H 1 , K j } = 0 . So, the addition theorems could help us to classify algebraica lly super in tegrable systems and vice versa we can get addition theorems by using known super in tegrable systems. 2 2 The action-angle v ariables for the op en lattices. In 197 5 Mo ser pro p osed construction of the action- a ngle v ar iables for the op en T o da la tt ice. This construction may b e extended o n other integrable systems related with the q uadratic r -matrix algebra s [24]. F or a ll these s ystems the a c t io n v ar iables I j are zero es of the p olynomial A ( λ ) = λ n + H 1 λ n − 1 + · · · H n = n Y j =1 ( λ − I j ) , { A ( λ ) , A ( µ ) } = 0 , (2.1) whose co efficien ts are indep enden t integrals of motion H 1 , . . . , H n in the inv olution. The ang le v ariables ω j are defined by the second p olynomial B ( λ ) having the s t a ndard Poisson brack et with the firs t p olynomial A ( µ ) [24] { B ( λ ) , A ( µ ) } = η λ − µ  B ( λ ) A ( µ ) − B ( µ ) A ( λ )  , η ∈ C , or { B ( λ ) , A ( µ ) } = η λ − µ  B ( λ ) A ( µ ) − B ( µ ) A ( λ )  + η λ + µ  B ( λ ) A ( µ ) + B ( − µ ) A ( λ )  . In this case angle v ariables ω j are eq ual to ω j = η − 1 ln B ( I j ) , j = 1 , . . . , n, η ∈ C , (2.2) Usually brack ets b et ween these tw o p olynomials are either par t of the Sklyanin a lgebra or the reflection equation algebra . It allows us to use the repre sen tation theor y in order to get single-v alued function A ( λ ) and B ( λ ) on M in initial physical v aria bles, see [24]. Using the angle v ariables ω j (2.2) we can obtain additiona l alg ebraic integrals of motion K j = e η ω j = B ( I j ) , { I k , K j } = 0 , k 6 = j, for these maximally sup erintegrable systems. Among suc h m a ximally sup erin tegr able s y stems we can find the op en T o da lattices asso ciated with the classical ro o t systems A n , B C n , and D n , the X X X Heisenberg magnets with boundarie s, and the discrete self-trapping mo del with b oundaries [24]. Remark 2 The prop erties of the tra jectories of motion is indep enden t on the choice of integrals of motion in the in volution I 1 , . . . , I n or H 1 , . . . , H n . This choice affects on the p ar ametrization only , i.e. on the explicit dep endence o f the co ordinates q j on the parameter t . Remark 3 The global action-a ng le v a r iables, the corresp onding Bir khoff co ordinates and the KAM- theorem for the p eriodic T o da lattices a re disc ussed in [9]. Up to now the co mpu ter ex periments indicate that p erio dic T o da lattice can not b e sup erin tegrable system on the whole phase spac e . 3 The action-angle v ariables for the St¨ ac k el systems. The system asso ciated with the name of St¨ ac kel [1 9 , 20] is a holono mic system on the phase space M = R 2 n , with the canonical v a riables q = ( q 1 , . . . , q n ) and p = ( p 1 , . . . , p n ): Ω = n X j =1 dp j ∧ dq j , { p j , q k } = δ j k . (3.1) The nondegenerate n × n St¨ ac kel ma t r ix S , whose j column dep ends on the co ordinate q j only , defines n functionally indep enden t integrals of mo tio n H k = n X j =1 ( S − 1 ) j k  p 2 j + U j ( q j )  . (3.2) 3 F r om this definition one immediately gets the s e pa rated relations p 2 j = n X k =1 H k S kj − U j ( q j ) (3.3) and the angle v ar ia bles ω i = X j =1 Z S ij d q j p j = X j =1 Z S ij d q j p P n k =1 H k S kj − U j ( q j ) . It allows r educing s o lution o f the equations o f motion to a problem in algebraic geo met r y [2 0 ]. Namely , let us supp ose that there ar e functions µ j and λ j on the canonical separa t ed v ariables µ j = u j ( q j ) p j , λ j = v j ( q j ) , { q i , p j } = δ ij , (3.4) which allows us to rewr ite separated equations (3.3) as equa tio ns defining the h yp erelliptic curves C j : µ 2 j = P j ( λ j ) ≡ u 2 j ( λ j ) n X k =1 H k S kj ( λ j ) − U j ( λ j ) ! , (3.5) where P j ( λ j ) are polynomials on λ j . In this case the action v ariables I k = H k (3.2) ha ve the canonical Poisson brackets (1 .1) with the a ngle v a riables ω i = n X j =1 Z A j S ij ( λ j ) p P j ( λ j ) d λ j = n X j =1 ϑ ij ( p j , q j ) , (3.6) which are usually the sums o f integrals ϑ ij of the first kind Ab elian differen tials on the hyperelliptic curves C j (3.5) [20, 24]. Remark 4 If the s eparated relations are obtained fro m the hyperelliptic cur v es by some subs t itution (3.4) the La grangian subma nifo ld of the corr esponding St¨ ack el sys tem is a direct pro duct of these hyperelliptic curves F = C 1 × C 2 × · · · × C n . If these curves are equa l C j = C then F may b e identified with the Jaco bian J ( C ) o f C [2 0 ]. 3.1 Addition theorems In generic case the action v ariables (3.6) are the sum of the multi-v alued functions ϑ ij . How ever, if w e are able to apply so me addition theo rem to the ca lculation of ω i (3.6) ω i = n X j =1 ϑ ij ( p j , q j ) = Θ i  K i  + const, (3.7) where Θ i is a multi-v alued function o n the alg ebraic argument K i ( p, q ), then one co uld get a dditional single-v alued integral of motion K i . T r eating λ n +1 as a function of the independent v a riables λ 1 , . . . , λ n the addition theorem (3.7) may be rewritten in the integral for m n X j =1 Z λ j S ij ( λ ) p P j ( λ ) d λ j − Z λ n +1 S ( λ ) p P n +1 ( λ ) d λ n +1 = const or in the differential for m ǫ 1 S i 1 ( λ 1 ) d λ 1 p P 1 ( λ 1 ) + · · · + ǫ n S in ( λ n ) d λ n p P n ( λ n ) − ǫ n +1 S ( λ n +1 ) d λ n +1 p P n +1 ( λ n +1 ) = 0 , ǫ i = ± 1 , (3.8) where choice of the ǫ i is r elated with the c hoice of the con tour s of integration. In this ca se additional int eg rals of mo tio n a re λ n +1 and S ( λ n +1 ) √ P n +1 ( λ n +1 ) , which hav e to b e sing le - v alued functions on the whole phase space. 4 Remark 5 If w e add the necessary words to the equation (3.8 ) at P i = P j , for all the i, j = 1 , . . . , n we o btain the famous Abel theorem, see as an example [3]. Recall, that Ab e l reduced equation (3 .8) to an a lgebraic ident ity (tra jectory o f motion), which is a conse quence of an expansion into par tia l fractions. Another standard form of the addition theo rem for the function f ( x ) lo oks like F  f ( x ) , f ( y ) , f ( x + y )  = 0 , ∀ x, y ∈ M , (3.9) where F is some function ass ociated with f . F or instance, we can show differen t forms (3.7 ) and (3.9) for the simplest addition theor em ln( x ) + ln( y ) = ln ( xy ) or e x e y = e x + y , (3.10) asso ciated with the zero -gen us hyp e relliptic curves C j C j : µ 2 j = P j ( λ j ) = f j λ 2 j + g j λ j + h j , j = 1 , . . . , n, (3.11) where f j , g j , h j are linea r functions on n integrals of motio n H 1 , . . . , H n . Namely , if the separated r e lations a re obtained from the zer o-gen us hyperelliptic c ur v es (3.11) and S ij = κ ij are constants we hav e ω i = n X j =1 ϑ ij = n X j =1 Z κ ij q f j λ 2 j + g j λ j + h j d λ = n X j =1 ln µ j + 2 f j λ j + g j 2 p f j ! κ ij f − 1 / 2 j = ln n Y j =1 µ j + 2 f j λ j + g j 2 p f j ! κ ij f − 1 / 2 j = ln K i . So, the angle v ariables ω i are lo g arithms as for the Moser sys t ems (2.2). Of course, if w e want to get algebraic or p o lynomial integrals of motion K i we hav e to impo s e some cons tr ains on κ ij and f j . W e hav e to underline that we use different h yp erelliptic curves C j in this exa m ple in contrast with the Euler example. 3.2 T ra ject or ies of motion According to the standard J acobi pro cedure of inv ersio n of the Abel map the v a riables q j are found from the equations n X j =1 Z S ij ( q j ) p P j ( q j , α 1 , . . . , α n ) d q j = β i , i = 1 , . . . , n, where β 1 = t and H j = α j . F rom these equatio ns one gets tra jectories in p ar ametric form q j ( t ), i.e. co ordinates q j are functions on the time β 1 = t and para m eter s α 1 , . . . , α n , β 2 . . . , β n . On the o ther hand, for the sup erin tegr able sys t ems if we substitute momenta fr om the separated equations (3.3) p j = v u u t n X k =1 α k S kj ( q j ) − U j ( q j ) int o additional a lgebraic integrals of motion K j one g e t s tra jector ies of the motion in the algebr aic form F j = K j ( q 1 , . . . , q n , α 1 , . . . , α n ) − k j = 0 , j = 1 , . . . , n − 1 . (3.12) 5 Remark 6 There are many different para meterizations of the algebra ic cur ves. Since, tra jectories may be a ssociated with the differ en t St¨ ack el systems (3 .12), which are related by the c a nonical trans- formation of the time t → e t : d e t = v ( q ) dt, v ( q ) = det S det e S . (3.13) Here S and e S are the St¨ ac kel matrices for the dual systems with common tra jectories [21, 23]. E xistence of the such dual systems is related with the fac t that the Abel map is surjective and g enerically injectiv e if n = g only . Of co urse, such dual St¨ ac kel s ystems hav e so-ca lled equiv alent metrics. 4 Logarithmic angle v ariables In or der to classify tw o-dimensiona l supe rin tegrable systems as s ociated with the addition theorem (3.10) we can start with a pair of the Riemann surfaces C j : µ 2 = P j ( λ ) = f λ 2 + g j λ + h j , j = 1 , 2 , (4.1) where f = αH 1 + β H 2 + γ , g j = α g j H 1 + β g j H 2 + γ g j , h j = α h j H 1 + β h j H 2 + γ h j , and α , β a nd γ are rea l or co mp lex num b ers. W e can use additio n theorem (3.1 0 ) if w e we fix last row of the St¨ ack el matrix o nly . N a mely , substituting S 2 j ( λ ) = κ j (4.2) int o the (3 .3-3.6 ) one gets ϑ 2 j = Z S 2 j ( q j ) d q j p j = Z κ j d λ j p P j = κ j f − 1 / 2 ln  µ j + 2 f λ j + g j 2 √ f  , (4.3) so the angle v ariable ω 2 = 1 √ f ln  p 1 u 1 + P ′ 1 2 √ f  κ 1  p 2 u 2 + P ′ 2 2 √ f  κ 2  , P ′ j = dP j ( λ ) dλ     λ = v j ( q j ) , is the m ulti-v alued function on the desir ed algebra ic a rgumen t K =  p 1 u 1 + P ′ 1 2 √ f  κ 1  p 2 u 2 + P ′ 2 2 √ f  κ 2 . If κ 1 , 2 are p ositive integer, then K =  1 2 √ f  κ 1 + κ 2  K ℓ + p f K m  (4.4) is the generating function o f p olynomial integrals of motion K m and K ℓ of the m -th and m ± 1 -th order in the momenta, resp ectively . As a n example we hav e K m = 2 ( p 1 u 1 P ′ 2 + p 2 u 2 P ′ 1 ) , κ 1 = 1 , κ 2 = 1 , K m = 2 P ′ 2  2 p 2 u 2 P ′ 1 + p 1 u 1 P ′ 2  + 8 p 1 u 1 p 2 2 u 2 2 f , κ 1 = 1 , κ 2 = 2 , K m = 2 P ′ 2 2  3 p 2 u 2 P ′ 1 + p 1 u 1 P ′ 2  + 8 p 2 2 u 2 2  p 2 u 2 P ′ 1 + 3 p 1 u 1 P ′ 2  f , κ 1 = 1 , κ 2 = 3 , (4.5) where m = 1 , 3 , m = 3 , 5 m = 3 , 7, b ecause P ′ 1 , 2 and f a re linear functions on H 1 , 2 , which are the second order p olynomials o n moment a . 6 The co rrespo nd ing express ions for the K ℓ lo ok like K ℓ = P ′ 1 P ′ 2 + 4 p 1 p 2 u 1 u 2 f , κ 1 = 1 , κ 2 = 1 , K ℓ = P ′ 1 P ′ 2 2 + 4 f p 2 u 2 ( p 2 u 2 P ′ 1 + 2 p 1 u 1 P ′ 2 ) , κ 1 = 1 , κ 2 = 2 , K ℓ = P ′ 1 P ′ 2 3 + 12 P ′ 2 p 2 u 2 ( P ′ 2 p 1 u 1 + P ′ 1 p 2 u 2 ) f + 1 6 p 1 u 1 p 3 2 u 3 2 f 2 . κ 1 = 1 , κ 2 = 3 . (4.6) Of course, we ca n try to get another p olynomial integrals of motion using the following recurrence relations K j − 1 = { K j , H 2 } , , a nd { H 1 , K j } = 0 , j = m, m ± 1 , m ± 2 , . . . . (4.7) see examples in [25]. Remind, tha t w e get p olynomial integrals of mo t io n K m and K ℓ for the sp e cial St¨ ac kel matrices only . The imp osed necessar y condition (4.2) leads to some r estrictions on the substitutions λ j = v j ( q j ) , µ j = u j ( q j ) p j . In fact, after these substitutions we o btain the following expre ssion for the St¨ ac kel matr ix S ( q ) =       αv 2 1 + α g 1 v 1 + α h 1 u 2 1 αv 2 2 + α g 2 v 2 + α h 2 u 2 2 β v 2 1 + β g 1 v 1 + β h 1 u 2 1 β v 2 2 + β g 2 v 2 + β h 2 u 2 2       , det S 6 = 0 . (4.8) So, for a g iv en κ 1 , 2 expressions for ϑ 2 j (4.3) yield t wo differ en tial e quations on functions u, v and parameters β : S 2 j ( q j ) = κ j v ′ j ( q j ) u j ( q j ) = ⇒ κ j u j v ′ j = β v 2 j + β g j v j + β h j , j = 1 , 2 . (4.9) There ar e equations o n u, v a nd parameters β b ecause we have to solve these equations in the fixed functional space [25]. Prop osition 1 If κ j 6 = 0 e quations (4.9) have the fol low ing thr e e monomial solut i ons I β = 0 , β h j = 0 , u j = q j , v j = q β g j κ j j , II β g j = 0 , β h j = 0 , u j = 1 , v j = − κ j ( β q j ) − 1 , II I β = 0 , β g j = 0 , u j = 1 , v j = κ − 1 j β h j q j , (4.10) up to c anonic al t ra nsformations. The fourth solution ( IV) is the c ombination of the first and thir d solutions for the differ ent j ’s. Below we mark all the sup erint eg rable systems by num ber of the s olution from this Pro position. If we supp o se that after p oin t tra nsformation x = z 1 ( q ) , y = z 2 ( q ) , p x = w 11 ( q ) p 1 + w 12 ( q ) p 2 , p y = w 21 ( q ) p 1 + w 22 ( q ) p 2 , (4.11) the kinetic energ y has so me fixed form T = X  S − 1  1 j p 2 j = g 11 ( x, y ) p 2 x + g 12 ( x, y ) p x p y + g 22 ( x, y ) p 2 y , (4.12) then we will get so me additional restrictio ns on the separ ated relations and the St¨ ac kel ma tr ix. 7 F o r instance, if we supp ose that T = X  S − 1  1 j p 2 j = p x p y , then one will get the following a lgebraic equations w 11 w 21 =  S − 1  11 , w 12 w 21 + w 11 w 22 = 0 , w 12 w 22 =  S − 1  12 (4.13) and the partial differ en tial equations { x, p x } = { y , p y } = 1 , { p x , y } = { p y , x } = { p x , p y } = 0 . (4.14) on parameters α and functions z 1 , 2 ( q 1 , q 2 ), w kj ( q 1 , q 2 ). The remaining free parameters γ , γ h j , γ g j determine the corresp onding p oten tial part of the Hamil- tonian V ( x, y ). In fact, s ince integrals H 1 , 2 is defined up to the triv ial shifts H k → H k + c k , our po ten tial V ( x, y ) dep ends on three arbitrar y parameters only . 4.1 Examples According to [2 5 ] in this s ection we consider sup erin tegr able systems on a complex Euclidean space E 2 ( C ) with the following Hamiltonian H 1 = p x p y + V Z ( x, y ) , (4.15) where subscript Z = I , I I , I I I or I V indicates on the t yp e of the solution (4.10). The passag e from the conforma l co ordinate system ( x, y ) to ano t her co ordinate systems can b e realized by using the Beltrami partial differential equa t io ns. Remark 7 Accor din g to [21, 23] the St ¨ ack el t r ansformations (3.13) r elate systems as s ociated with the same solution o f (4.10) be c ause the corresp onding St¨ ack el matr ic e s S Z ( q ) are differed on the fir st row only . Example 1 Let us put κ 1 = 1 and κ 2 = 1 in (4.9) and try to solve equa t io ns (4 .13) - (4 .14). Her e is one system with firs t or der additional int eg ral K m (4.5) V (1) I I I = γ 1 xy + γ 2 ( x + y ) + γ 3 ( x − y ) , and s ev en systems with cubic integral of mo tion K m (4.5) V (1) I = γ 1 √ xy + γ 2 ( y − x ) 2 + γ 3 ( y + x ) √ xy ( y − x ) 2 , V (2) I = γ 1 xy + γ 2 ( y − x ) 2 + γ 3 ( y + x ) 2 , V (1) I I = γ 1 p y ( x − 1) + γ 2 p y ( x + 1) + γ 3 x √ x 2 − 1 , V (2) I I = γ 1 xy + γ 2 y 2 x 2 + 1 √ x 2 + 1 + γ 3 x √ x 2 + 1 , V (2) I I I = γ 1 √ xy + γ 2 √ x + γ 3 √ y , V (3) I I I = γ 1 y − 1 / 2 + γ 2 xy − 1 / 2 + γ 3 x , V I V = γ 1 ( y + x ) 2 + γ 2 ( y − x ) + γ 3 (3 y − x )( y − 3 x ) 3 , Solution of the recur rence relations (4 .7) K m − 1 = ±  2 p 1 p 2 u 1 u 2 + 2 v 1 v 2 f + v 1 g 2 + v 2 g 1  (4.16) is the additional polynomial in tegr a l of motion, whic h is the first o rder p olynomial for the s ystem with po ten tial V (1) I I I and the seco nd order p olynomial in the momenta for a nother systems. All these systems are listed in the Drach papers [7]. 8 Example 2 At κ 1 = 1 and κ 2 = 2 there is o ne sup erin tegr able system with c ubic additional integral K m (4.5) and quadra tic in teg ral K ℓ (4.6) V I I I = γ 1 (3 x + y )( x + 3 y ) + γ 2 ( x + y ) + γ 3 ( x − y ) , and s ev en systems with the rea l po tentials V I = γ 1 (3 x + y )( x + 3 y ) + γ 2 ( x + y ) 2 + γ 3 ( x − y ) 2 , V (1) I I = γ 1 xy + γ 2 p x 3 y + γ 3 x 2 , V (2) I I = γ 1 √ xy + γ 2 x 2 + γ 3 y x 3 , V (3) I I = γ 1 √ xy + γ 2 p x 3 y + γ 3 x 5 / 4 y 3 / 4 , V (4) I I = γ 1 xy + γ 2 y x 3 + γ 3 y 3 x 5 , V (1) I V = γ 1 √ xy + γ 2 ( √ x − √ y ) √ xy + γ 3 √ xy ( √ x + √ y ) 2 , V (2) I V = γ 1 xy + γ 2 ( x − y ) + γ 3 ( x + y ) 2 , for which integrals of mo t io n K m and K ℓ (4.5-4.6) ar e fifth and sixth o rder p olynomials in the momenta. Solution of the recur rence relations (4 .7 ) K m − 1 = { K m , H 2 } = 4 p 1 p 2 u 1 u 2 (2 f v 2 + g 2 ) + 4 v 2 (2 f v 1 + g 1 )( f v 2 + g 2 ) + (4 f h 2 + g 2 2 ) v 1 = 4 µ 2 ( µ 1 P ′ 2 + P ′ 1 ) − (4 f h 2 − g 2 2 ) λ 1 − 4 h 2 g 1 . is the additional polynomia l integral of motion, which is the s econd or der p olynomial for the system with p oten tial V I I I and the fourth order po lynomial in the mo menta fo r a nother systems. Solution of the next recurrence r elation K m − 2 = { K m − 1 , H 2 } is the ra tional function in mo men ta. Example 3 Now we present some sup erintegrable St¨ ac kel systems at κ 1 = 1 and κ 2 = 3. Here is one system with cubic additional integral K m (4.5) V (1) I I I = γ 1 ( x + 2 y )(2 x + y ) + γ 2 ( x + 2 y ) + γ 3 (2 x + y ) , S = ( a b 1 1 ) , and s ev en systems with the rea l po tentials V I = γ 1 ( x + 2 y )(2 x + y ) + γ 2 ( x + y ) 2 + γ 3 ( x − y ) 2 , V (1) I I = γ 1 xy + γ 2 x 2 / 3 y 4 / 3 + γ 3 x 1 / 3 y 5 / 3 , V (2) I I = γ 1 √ xy + γ 2 √ y x 5 / 2 + γ 3 y 2 x 4 , V (3) I I = γ 1 √ xy + γ 2 x 4 / 3 y 2 / 3 + γ 3 x 7 / 6 y 5 / 6 , V (4) I I = γ 1 xy + γ 2 y 2 x 4 + γ 3 y 5 x 7 , V (2) I I I = γ 1 ( x 2 − 5 x √ y + 4 y ) + γ 2 x √ y + γ 3 √ y , V I V = γ 1 ( x + 5 y )(5 x + y ) + γ 2 ( x − y ) + γ 3 ( x − y ) 2 , for which int eg rals of mo tion K m and K ℓ (4.5-4.6) are seven th and eig h ts order p olynomials in the momenta. 9 4.2 Algebra of in tegrals of motion F o r a ll the superintegrable systems co nsidered in the previous s ection the algebra of in tegrals of motion H 1 , 2 and K m , K ℓ (4.5) is the po lynomial algebr a in terms of the co efficient s of the h yp erelliptic curves { H 2 , K m } = 2 K ℓ , { H 2 , K ℓ } = 2 f K m , { K m , K ℓ } = ± Φ Z , where p olynomial F Z depe nds on the type of solutio n (4.1 0 ) o nly: Φ I = 2(4 f h 2 − g 2 2 ) κ 2 − κ 1  4 f ( κ 2 1 h 2 g 1 + κ 2 2 h 1 g 2 ) − g 1 g 2 ( κ 2 2 g 1 + κ 2 1 g 2 )  , Φ I I = 4(4 f h 2 − g 2 2 ) κ 2 − κ 1  4 f ( κ 2 + κ 1 ) h 2 h 1 − κ 1 h 1 g 2 2 − κ 2 h 2 g 2 1  ∓ K ν m , Φ I I I = 4(4 f h 2 − g 2 2 ) κ 2 − κ 1  4 f ( κ 1 h 2 + κ 2 h 1 ) − κ 1 g 2 2 − κ 2 g 2 1  f , (4.17) Φ I V = 2(4 f h 2 − g 2 2 ) κ 2 − κ 1  4 f (2 κ 2 f h 1 − κ 1 h 2 g 1 ) − 2 κ 2 f g 2 1 + κ 1 g 1 g 2 2  . Here ν = 1 for κ 1 = κ 2 and ν = 2 for κ 1 6 = κ 2 . Cho ice of the sign + or − dep ends on κ ’s to o. The St¨ ack el transfo rmations ( 3 .13) relate sys t ems asso ciated with one type of the solutions (4.10), whereas algebra of integrals o f motion is inv ariant with r espect to such transfor mations. 5 Elliptic angle v ariables The first demonstration of the existence of an addition theor em for elliptic functions is due to E uler [8], who show ed that the differential rela tion κ 1 dx √ X + κ 2 dy √ Y = 0 (5.1) connecting the most genera l quartic function of a v ariable x X = a x 4 + 4 b x 3 + 6 c x 2 + 4 d x + e (5.2) and the same function Y o f another v ar iable y, leads to an alg ebraical relation b et ween x and y, X and Y at integer κ 1 , 2 . As a n example at κ 1 , 2 = 1 one gets √ X − √ Y x − y ! 2 = a (x + y) 2 + 4 b (x + y) + C where C is the a r bitrary co nstan t of in tegra tion. This a lgebraic relation when rationa lized leads to a symmetrical biquadra tic form o f x and y F (x , y) = a x 2 y 2 + 2 b xy(x + y ) + c (x 2 + 4 xy + y 2 ) + 2 d (x + y) + e = 0 , (5.3) which defines the conic se c t io n on the plane (x , y), which then will b e classica l t r a jectory of motion in the configura t io nal space. According to [3] we co uld replace the constant C in the Euler integral re la tion b y 4 c + 4 s , where s = F (x , y) − √ X √ Y 2(x − y ) 2 = 1 4 √ X − √ Y x − y ! 2 − (x + y) 2 4 − b (x + y) − c , (5.4) is the famous algebra ic Euler integral. T reating s as a function of the indep enden t v aria bles x a nd y, one gets the following addition theor em dx √ X + dy √ Y + d s √ S = 0 (5.5) 10 Of co urse, the Euler addition theorem is a v ery specia l ca se of the Ab el theor em [3]. As an example, the similar explicit a dd itio n for mulas for genus t wo hyper elliptic curves may b e found in [6]. The p olynomial S in (5.5) may b e defined in the algebraic fo rm √ S =  Y 1 x + Y 2 ) √ X −  X 1 y + X 2  √ Y (x − y) 3 , (5.6) where X 1 = ( a x 3 + 3 b x 2 + 3 c x + d ) , X 2 = b x 3 + 3 c x 2 + 3 d x + e and s imilar to Y 1 , 2 , or in the W eierstrass form S = 4 s 3 − g 2 s − g 3 (5.7) where g 2 = ae − 4 bd + 3 c 2 , g 3 = ace + 2 bcd − ad 2 − e b 2 − c 3 , (5.8) are the quadr iv ar ian t and cubicv ariant of the quar tic X [2], resp ectiv ely . F o r κ 1 = 1 and κ 2 = 2 we present Euler’s equa tion of the tra jectory of motion at c = d = 0 only: F (x , y) = ( e − a y 4 )x + 2y p e ( a y 4 + 6 c y 2 + e ) , see page 355 in [8]. 5.1 Classification of the Euler sup erin tegrable systems According to [11], in order to use the Euler addition theorem (5 .5 ) for construction of the sup erin te- grable St¨ ac kel systems we have to s t a rt with the genus one hypere llipt ic curv e µ 2 = P ( λ ) , where P (x) = X , and a pair of arbitrar y substitutions λ j = v j ( q j ) µ j = u j ( q j ) p j , j = 1 , 2 , where p and q ar e canonica l v ar iables { p j , q i } = δ ij . This hype relliptic curve and subs t itutions give us a pa ir o f the separated rela tions p 2 j u 2 j ( q j ) = av j ( q j ) 4 + 4 bv j ( q j ) 3 + 6 cv j ( q j ) 2 + 4 dv j ( q j ) + e, j = 1 , 2 , (5.9) where co efficien ts a, b, c, d and e of the qua rtic X (5 .2) are linear functions of integrals o f motion H 1 , 2 : a = α 1 H 1 + α 2 H 2 + α, b = β 1 H 1 + β 2 H 2 + β , . . . , e = ǫ 1 H 1 + ǫ 2 H 2 + ǫ . As ab ov e the separa ted relations (5.9) coincide with the Ja cobi rela tions for the uniform St¨ ack el sys t ems [19, 20, 21] p j = v u u t n X k =1 H k S kj − U j ( q j ) (5.10) where S is the so-c alled St¨ ack el matrix and U j is the St¨ ack el p o ten tial: S ij = u − 2 j ( α i v 4 j + 4 β i v 3 1 + 6 γ i v 2 j + 4 δ i v j + ǫ i ) , U j = u − 2 j ( αv 4 j + 4 β v 3 1 + 6 γ v 2 j + 4 δv j + ǫ ) , i, j = 1 , 2 . Solving these separ ated rela tio ns (5.9-3.3) with r espect to H 1 , 2 one gets pair of the St¨ ackel integrals of motion in the inv olutio n H k = 2 X j =1 ( S − 1 ) j k  p 2 j + U j ( q j )  , k = 1 , 2 , (5.11) 11 and a ngle v ariables ω i = 1 2 2 X j =1 Z S ij d q j p j = 1 2 2 X j =1 Z S ij d q j p P n k =1 H k S kj − U j ( q j ) (5.12) canonically conjugated with the actio n v ar iables H 1 , 2 . In generic case the action v a riables (5.12) ar e the sum of the multi-v alued functions. How ever, if we are able to apply so m e addition theor em to the calcula t io n of ω 2 ω 2 = 1 2 Z v 1 ( q 1 ) S 21 ( λ )d λ p P ( λ ) + 1 2 Z v 2 ( q 2 ) S 22 ( λ )d λ p P ( λ ) = 1 2 Z s d s √ S , then one could get additiona l single-v alued int eg rals of mo tion s and S : { H 1 , ω 2 } = 0 ⇒ { H 1 , s } = { H 1 , S } = 0 . Of course, opp ortunit y to apply some addition theorem to computation of the angle v aria ble ω 2 and of the additional single-v a lued integrals of motio n s, S leads to some r estrictions on o ur qua rtic P ( λ ) and s ubstit utions λ j = v j ( q j ) , µ j = u j ( q j ) p j , see [11]. F o r instance, if we wan t to use the Euler a ddit io n theorem (5.5) we hav e to put S 21 ( λ ) = 1 , S 22 ( λ ) = ± 1 . Such as S ij ( q ) = v ′ j ( q j ) u j ( q j ) S ij ( λ ) these restric t io ns are eq uiv alent to the following equations κ j u j v ′ j = α 2 v 4 j + 4 β 2 v 3 j + 6 γ 2 v 2 j + 4 δ 2 v j + ǫ 2 , κ 1 = 1 , κ 2 = ± 1 , (5.13) on functions u ( q ) , v ( q ) and co efficients α 2 , β 2 , . . . , ǫ 2 of the qua rtic, b ecause we have to so lv e these equations in some fixed functional s pace [25]. It’s ea sy to prov e that there ar e fiv e monomial so lut io ns I u j = 1 , v j = q j , ǫ 2 6 = 0 II u j = q j , v j = q 4 δ 2 κ − 1 j j , δ 2 6 = 0 , II I u j = 1 , v j = q − 1 j , γ 2 6 = 0 IV u j = q − 1 j , v j = q − 1 j , β 2 6 = 0 , V u j = q − 2 j , v j = q − 1 j , α 2 6 = 0 (5.14) up to canonical transformatio ns of the s eparated v ariables ( p j , q j ) and transformations of integrals o f motion H j → σ H j + ρ . Here notatio ns α 2 6 = 0, β 2 6 = 0 , . . . mean that other para met er s are equal to zero. As above, if w e supp ose that after some po in t transformation (4.1 1 ) kinetic part of the Hamilton function ha s a sp ecial form (4.12), then one g ets some additional r estrictions (4 .13-4.14) ) on the co efficien ts of the quar tic P ( λ ). The remaining free parameters α, . . . , ǫ determine p oten tial part of the Hamiltonian V ( x, y ). In fact, since int eg rals H 1 , 2 is de fined up to the trivial shifts H k → H k + ρ k , our p oten tial V ( x, y ) depends on three arbitr ary parameters o nly . Summing up, we hav e prov ed that classification of the Euler sup erint eg rable systems on the plain is e quiv alent to solution of the eq uations (5.13,4.13,4.14). F or a ll the p ossible so lutions a ddit io nal quadratic in momenta integral o f motion lo oks like K 2 = s + c = F ( v 1 , v 2 ) − p 1 u 1 p 2 u 2 2( v 1 − v 2 ) 2 + c , 12 see (5.4) and additional cubic in momen ta integral of mo t io n is eq ua l to K 3 = √ S ≡ p 4 s 3 − g 2 s − g 3 see (5.6) and (5.7). Remark 8 Of cour se, additional integrals of motion for t he n - dimens ional sup erin tegra ble systems are related with the Abel theorem as well. As an example w e present the quadratic in momenta Richelot int eg ral [18] s = λ 2 1 . . . λ 2 n   n X j =1 µ j λ 2 j F ′ ( λ j )   2 − a 1  1 λ 1 + · · · + 1 λ n  − a 0  1 λ 1 + · · · + 1 λ n  2 for the uniform St¨ ack el sy s tems linearized on the J acobian of the following hypere lliptic curve µ 2 = a 2 n λ 2 n + · · · + a 1 λ + a 0 , such tha t the St¨ a c kel matrix is a low est block of the corresp onding Brill-No ether matrix [20, 21]. Here F( λ ) = Q ( λ − λ j ) and λ j = v j ( q j ) , µ j = u j ( q j ) p j are the suitable substitutions. 5.2 Examples Let us find all the E uler super in tegrable systems on a complex Euc lide a n space E 2 ( C ) H 1 = p x p y + V ( x, y ) (5.15) with the r eal p oten tials V . Solving equations (5.1 3 ,4.1 3 ,4 .14 ) one gets the following five sup erint eg rable po ten tials V 1 = α ( x + y ) + β ( y + 3 x ) x − 1 / 2 + γ x − 1 / 2 , V 2 = αy ( x + y 2 ) + β ( x + 3 y 2 ) + γ y , V 3 = α ( x + 3 y )(3 x + y ) + β ( x + y ) + γ ( x − y ) 2 V 4 = αxy − 3 − β y − 2 − γ xy , V 5 = α x 2 + β x 3 / 2 √ y − 1 + γ x 1 / 2 √ y − 1 . (5.16) Recall, tha t implicitly all these systems hav e b e e n found by Euler in 1761 [8]. Poten tials V 1 and V 3 in explicit form hav e b een found by Dra c h [7], the ( ℓ ) and ( g ) ca ses, wher e a s p oten tial V 2 , V 4 and V 5 may b e found in [1 4 ], a s the E 10 , E 8 and E 17 cases, r espectively . The first and fifth solutions (5.14) of the eq ua tions (5.13) are related with p oten tials V 1 , 2 . The second a nd fourth solutions give us p oten tial V 3 , wher eas third solution yields p oten tials V 4 , 5 . F or brevity we present the main s tages of calculatio ns only , all the details may b e found in [11]. Case 1 If we take fir st solution from (5.14) u j = ± 1 , v j = ± q j , ⇒ p 2 1 = P ( q 1 ) , ( − p 2 ) 2 = P ( − q 2 ) and q uartic P ( λ ) = − α 2 λ 4 + 2 β λ 3 + H 1 λ 2 + 2 γ λ + H 2 , 13 then after the following change of v ar iables x = ( q 1 − q 2 ) 2 4 , p x = p 1 − p 2 q 1 − q 2 , y = ( q 1 + q 2 ) 2 2 , p y = p 1 + p 2 q 1 + q 2 , (5.17) we obtain the St¨ a c kel in tegr als H 1 = p x p y + α ( x + y ) + β (3 x + y ) √ x + γ √ x , H 2 = ( p x − p y )( p x x − p y y ) − α ( x − y ) 2 2 − β ( x − y ) 2 √ x + γ ( x − y ) √ x , the Euler integrals of motion K 2 = s + c = p 2 y 4 + αx 2 + β √ x, K 3 = √ S = p 2 y ( p x − p y ) 4 + α ( p x − p y ) x 2 + β (2 p x x − 3 p y x + p y y ) 4 √ x + γ py 4 √ x . (5.18) The same system may b e o bt a ined b y using fifth substitution from (5 .14) . Case 2 Using the sa me first solution (5.1 4) a nd another quartic P ( λ ) = − α 4 λ 4 − β λ 3 − γ 2 λ 2 + H 1 λ + H 2 after the following change of co ordinates x = ( q 1 + q 2 ) 2 4 , p x = p 1 + p 2 q 1 + q 2 , y = q 1 − q 2 2 , p y = p 1 − p 2 , we can get sup erin tegr able St¨ ack el system H 1 = p x p y + αy ( x + y 2 ) + β ( x + 3 y 2 ) + γ y , H 2 = p 2 y 4 + p 2 x x − y p y p x + α (3 y 2 + x )( x − y 2 ) 4 + 2 β y ( x − y 2 ) + γ 2 ( x − y 2 ) , with the quadra t ic Euler integral K 2 = s + c = p 2 x 4 + αy 2 4 + β y 2 and w ith the cubic in teg ral of motion K 3 = √ S = p 3 x 4 + α  3 p x ( x + y 2 ) − p y y  8 + β (6 p x y − py ) 8 + γ p x 8 . The same system may b e o bt a ined b y using fifth substitution from (5 .14) as well. Case 3 If we take the second so lut io n from (5 .14) u j = q j , v j = q 2 j , ⇒ p 2 1 q 2 1 = P ( q 2 1 ) , p 2 2 q 2 2 = P ( q 2 2 ) and q uartic P ( λ ) = − αλ 4 − β 2 λ 3 + H 1 λ 2 + H 2 λ + γ , 14 after canonical trans f o r mations (5.17) we will get the follo wing sup erin tegr able St¨ ack el system H 1 = p x p y + α ( x + 3 y )(3 x + y ) + β ( x + y ) + γ ( x − y ) 2 , H 2 = ( p x − p y )( p x x − p y y ) − 2 α ( x + y )( x − y ) 2 − β ( x − y ) 2 2 − 2 γ ( x + y ) ( x − y ) 2 . Using the Euler addition theo rem (5.5) we can get the E ule r integral K 2 = s + c = ( p x + p y ) 2 16 + α ( x + y ) 2 + β 4 ( x + y ) and the cubic in momen ta integral of mo tio n K 3 = √ S = ( p x − p y ) 2 ( p x + p y ) 32 − α ( x − y )  p x (5 x + 3 y ) − (3 x + 5 y ) p y  8 − β ( p x − p y )( x − y ) 16 − γ ( p x + p y ) 8( x − y ) 2 . The same system may b e o bt a ined b y using four th substitution from (5 .1 4 ). Case 4 If we take the third s o lution from (5.14) u j = ± 1 , v j = ± q − 1 j , ⇒ p 2 1 = X ( q − 1 1 ) , p 2 2 = Y ( − q − 1 2 ) and q uartic P ( λ ) = αλ 4 − β λ 3 + H 2 λ 2 + H 1 λ − γ , then after canonical tra nsformation x = √ q 1 q 2 , p x = p 1 q 1 + q 2 p 2 √ q 1 q 2 , y = q 1 − q 2 √ q 1 q 2 , p y = √ q 1 q 2 ( p 1 q 1 − q 2 p 2 ) q 1 + q 2 we will get the following sup erint eg rable St¨ ack el system H 1 = p x p y + αy x 3 + β x 2 + γ xy , H 2 = p 2 y + ( p x x − y p y ) 2 4 − α ( y 2 + 1 ) x 2 − β y x + γ x 2 . Using the Euler addition theo rem (5.5) one gets additional quadr atic in momenta Euler integral K 2 = s + C = ( p x x − y p y ) 2 16 − αy 2 4 x 2 − β y 4 x and the cubic integral of motion K 3 = √ S = p 2 y ( p x x − y p y ) 8 + α (3 y p y + xp x ) 8 x 2 + β p y 4 x + γ ( p x x − y p y ) x 2 8 . Case 5 Using the sa me third solution (5.14) and ano t her q ua rtic P ( λ ) = 4 αλ 4 + 4 β λ 3 + H 2 λ 2 + 2 γ λ + H 1 after the following change of v aria bles x = q 1 q 2 2 , p x = p 1 q 1 + q 2 p 2 q 1 q 2 , y = q 2 1 + q 2 2 2 q 1 q 2 , p y = q 1 q 2 ( p 1 q 1 − q 2 p 2 ) q 2 1 − q 2 2 , we can get more complica ted super in tegrable system with the St¨ a ckel integrals of motion H 1 = p x p y + α x 2 + β x 3 / 2 √ y − 1 + γ x 1 / 2 √ y − 1 , H 2 = ( xp x − p y − p y y )( xp x + p y − p y y ) − 4 αy x − 2 β (2 y − 1) x 1 / 2 √ y − 1 − 2 γ x 1 / 2 √ y − 1 . 15 The Euler integral of motion is eq ual to K 2 = s + c = ( p x x + p y − p y y ) 2 4 − α ( y − 1) x − β √ y − 1 √ x , whereas the cubic in momenta integral r eads as K 3 = √ S = − p y ( xp x + p y − p y y ) 2 2 + 2 α ( y − 1) p y x − β ( p x x − 3 p y y + 3 py ) 2 x 1 / 2 √ y − 1 − γ x 1 / 2 ( p x x + p y − p y y ) 2 √ y − 1 . This lis t of exa mples may be ea sily broadened b ecause Euler prop osed constructio n of the a lgebraic int eg rals o f motion for the equatio ns (5.1) with a n y integer κ ’s. W e discuss the E uler construction of the algebra ic integrals of mo tion for s uperintegrable systems at κ 1 , 2 = ± 1 only . O f cour se, the most int er esting case is the case with the different κ 1 , 2 in (5.1) for which the corr esponding additional int eg rals of motion will b e hig her order p olynomials in momen ta as for log arithmic angle v a riables. 5.3 The quadratic in tegrals of motion It is easy to prove that the algebra of integrals of motion H 1 , 2 and K 2 is the quadratic Poisson algebra bec ause { H 2 , K 2 } = σK 3 = σ p S ( s ) , wher e    σ = 2 , V 1 , σ = − 2 , V 2 , V 4 , V 5 , σ = 4 , V 3 , and { H 2 { H 2 , K 2 }} = { H 2 , σ √ S } = σ 2 2 S ′ = σ 2 2  12 s 2 − g 2  = Φ( H 1 , H 2 , K 2 ) , (5.19) where Φ( H 1 , H 2 , K 2 ) is the second order p olynomial such as s = K 2 − c and g 2 is quadr iv ar ian t of the quartic (5.8 ). Another details on the quadratic Poisson algebr as of int eg rals of motion may b e found in [5]. The sear c h of the tw o dimensio nal manifolds who se the geo desics are curves whic h p ossess tw o additional quadr atic in tegr als of motion was initiated by Darb oux [4], who found fiv e c lasses of the metrics. These metrics or ”formes essen tielles” are tabulated in ” T ableau” b y Ko enigs [16] and in [15]. The s up erintegrable Darb oux-Ko enigs systems hav e a generic conforma l Hamiltonian H 1 = p ξ p η g( ξ , η ) + V ( ξ , η ) , where the Darb oux-Ko enigs metric g is a metric o n the Liouville surface [4, 5] if g( ξ , η ) = F ( ξ + η ) + G ( ξ − η ) , and K 2 = p ξ 2 + p η 2 − 2 p ξ p η β ( ξ , η ) g( ξ , η ) + Q ( ξ, η ) or metric g is a metric on the Lie surface if g( ξ , η ) = ξ F ( η ) + G ( η ) , and K 2 = p ξ 2 − 2 p ξ p η β ( ξ , η ) g( ξ , η ) + Q ( ξ, η ) . Super in tegrable systems a ssociated with the Liouv ille surfaces are separa ble in the tw o different o r- thogonal systems of coor dinates. It means that t wo pairs of in tegrals of motion ( H 1 , H 2 ) and ( H 1 , K 2 ) take on the St¨ ackel form (3.2) after some different p oin t tra nsformations (4.11). F o r the superintegrable systems a s sociated with the Lie surfaces only one pair of in tegr als ( H 1 , H 2 ) may be reduced to the St¨ a c kel form (3.2), w her eas second pair of integrals ( H 1 , K 2 ) do esn’t sepa rable in the class of the p oin t tr a nsformations (4.11). It’s ea sy to prove that tw o systems with p oten tia ls V 1 and V 2 are defined on the Lie surfaces. The remaining systems with po ten tials V 3 , V 4 and V 5 are defined on the Liouv ille surfaces. The second 16 separated v ar iables e q 1 , 2 for in teg rals of motion H 1 , K 2 may b e found b y using the softw are prop osed in [10]: x = e q 2 1 − e q 2 2 4 , y = e q 2 1 + e q 2 2 4 , for V 3 , x = e q 2 e q − 2 1 , y = e q 2 e q 2 1 , for V 4 , V 5 . It is easy to pr ove that the co r responding separ ated r elations e p 2 j = P ( e q j ) define tw o different zer o-gen us hyperelliptic curves and lead to lo garithmic angle v aria bles [25]. Thus, for these thr e e Euler sup er- int eg rable systems there ar e tw o different addition theorems: additio n theorem for elliptic functions (5.5) and a ddit io n theo r em for logarithms ln x + ln y = ln xy. So, multiseparability of the sup erin- tegrable systems may be asso ciated with o ccurrence of the different addition theorems for a given sup e rin tegrable hamiltonian. According to [13], the St¨ ack el integrals of motio n are in the bi-inv olutio n with resp ect to a pair of compatible Poisson brackets { H i , H j } = { H i , H j } ′ = 0 , where the second Poisson tenso r P ′ = N P is obtained by the fo llowing r ecursion op erator N : N ∂ ∂ q k = n X i =1 L i k ∂ ∂ q i + X ij p j ∂ L j i ∂ q k − ∂ L j k ∂ q i ! ∂ ∂ p i , N ∂ ∂ p k = n X i =1 L k i ∂ ∂ p i . Here L is so-ca lled Benenti tenso r [1]. The r e is symbolic softw are [10], whic h allows us to o btain this tensor starting with the Ha milt o nia n H 1 only . So, it is easy to prove that a n y sup erin tegra ble s ystem asso ciated with the Liouville surfa ce has t wo differen t linear in momen ta Poisson bivectors P ′ and P ′′ , whic h are compatible with the canonical bivector P a nd incompatible to each other. On the o t her hand, sup erin tegr able system ass ociated with the Lie surface has one linear Poisson biv ector P ′ only . It will be interesting to understand the geometric origin of this phenomena. Moreov er , we can find s o me integrable sy stems asso ciated with the Lie surfaces, which do esn’t separable in the class of the point transformations and whic h hasn’t linear second Poisson bivectors. As an example, we present the following int eg rable system with the qua dratic in tegr als of motion H 1 = p x p y + αy − 3 / 2 + β xy − 3 / 2 + γ x, K 2 = p 2 x − 4 β y 1 / 2 + 2 γ y , which hasn’t any other quadratic integrals. It will b e interesting to class if y such systems and describ e the non-linear bi-hamiltonian structur es asso ciated with the Lie surfac e s. The r esearch w as pa rtially supp orted by the RFBR gra n t 06-0 1-00140. References [1] S. B e ne nti, Int ri n s ic char acterization of the variable sep ar ation in the Hamilton-J a c obi e quation , J. Math. Phys. v.38, pp.657 8-6602, 1997. [2] W.S. Bur nside and A.W. Pan ton, The ory of Equations , Longmans, 188 6. [3] A. Cayley , An Elementary T r e atise on El liptic F unctions , Lo ndon, 1876 . A.G. Greenhill, The applic ations of el liptic functions , London, 189 2. [4] G. Dar boux, Le¸ cons sur la Th ´ eorie G´ e n´ er ale des Surfaces, 18 98. [5] C. Dask aloy annis, K. 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