Pairs of commuting Hamiltonians quadratic in momenta

P airs of omm uting Hamiltonians, quadrati in momen ta V.G. Marikhin 1 and V.V. Sok olo v 1 1 L.D. Landau Institute for Theoretial Ph ysis RAS, Moso w, Russia In the ase of t w o degree system the pairs of quadrati in momen ta Hamiltonians omm uting aording the standard P oisson bra k et are onsidered. The new man y-parametrial families of su h pairs are founded. The univ ersal metho d of onstruting the full solution of Hamilton - Jaobi equation in terms of in tegrals on some algebrai urv e is prop osed. F or some examples this urv e is non-h yp erellipti o v ering o v er the ellipti urv e. MSC n um b ers: 17B80, 17B63, 32L81, 14H70 A ddress : Landau Institute for Theoretial Ph ysis RAS, K osygina st.2 ,Moso w, Russia, 119334, E-mail : m vgitp.a.ru, sok olo vitp.a.ru 1 1 P airs of quadrati hamiltonians In pap ers [1, 2, 3, 4 , 5 , 6, 7℄ the problem of the omm uting pairs of Hamiltonians quadrati in momen ta w as onsidered. Consider pair of Hamiltonians in the form H = ap 2 1 + 2 bp 1 p 2 + cp 2 2 + dp 1 + ep 2 + f , (1.1) K = Ap 2 1 + 2 B p 1 p 2 + C p 2 2 + D p 1 + E p 2 + F , (1.2) omm uting with resp et to standart p oisson bra k et { p α , q β } = δ αβ . The o eien ts in form ulas (1.1 ),(1.2) - some (lo ally) analitial funtions of the v ariables q 1 , q 2 . Theorem 1. A ny p airs of  ommuting Hamiltonians (1.1 )-(1.2 )  an b e  anoni al ly tr ansforme d by ˆ P 1 = P 1 + ∂ F ( s 1 , s 2 ) ∂ s 1 , ˆ P 2 = P 2 + ∂ F ( s 1 , s 2 ) ∂ sq 2 to the p air of the form H = U 1 − U 2 s 1 − s 2 , K = s 2 U 1 − s 1 U 2 s 1 − s 2 , (1.3) wher e U 1 = S 1 ( s 1 ) P 2 1 + p S 1 ( s 1 ) S 2 ( s 2 ) Z s 1 ( s 1 − s 2 ) P 2 − S 1 ( s 1 ) Z 2 s 1 4( s 1 − s 2 ) 2 + V 1 ( s 1 , s 2 ) , U 2 = S 2 ( s 2 ) P 2 2 − p S 1 ( s 1 ) S 2 ( s 2 ) Z s 2 ( s 1 − s 2 ) P 1 − S 2 ( s 2 ) Z 2 s 2 4( s 2 − s 1 ) 2 + V 2 ( s 1 , s 2 ) , (1.4) V 1 = 1 2 p S 1 ( s 1 ) ∂ q 1  p S 1 ( s 1 )) Z 2 s 1 s 1 − s 2  + f 1 ( s 1 ) , V 2 = 1 2 p S 2 ( s 2 ) ∂ q 2  p S 2 ( s 2 )) Z 2 s 2 s 2 − s 1  + f 2 ( s 2 ) (1.5) for some funtions Z ( s 1 , s 2 ) , S i ( s i ) and f i ( s i ) . Poisson br aket { H , K } e quals to zer o if and only if Z s 1 ,s 2 = Z s 1 − Z s 2 2( s 2 − s 1 ) (1.6) and  Z s 1 ∂ ∂ s 2 − Z s 2 ∂ ∂ s 1   V 1 − V 2 s 1 − s 2  = 0 . (1.7) Pro of W e in tro due new o ordinates s 1 , s 2 , su h that the quadrati parts of H , K (1.1,1.2 ) are diagonal: Let s 1 , s 2 b e the ro ots of equations Φ( s, q 1 , q 2 ) = ( B − bs ) 2 − ( A − as )( C − cs ) = 0 , (1.8) Then the anonial transformation ( q 1 , q 2 , p 1 , p 2 ) → ( s 1 , s 2 , P 1 , P 2 ) : p 1 = − ( Φ 1 q 1 Φ 1 s 1 P 1 + Φ 2 q 1 Φ 2 s 2 P 2 ) , p 2 = − ( Φ 1 q 2 Φ 1 s 1 P 1 + Φ 2 q 2 Φ 2 s 2 P 2 ) , (1.9) 2 where Φ i = Φ( s i , q 1 , q 2 ) under onditions { H , K } = 0 transforms pairs (1.1 ),(1.2) to the form H = U 1 − U 2 s 1 − s 2 , K = s 2 U 1 − s 1 U 2 s 1 − s 2 , (1.10) where U 1 = S 1 ( s 1 ) P 2 1 + ˜ dP 1 + ˜ eP 2 + ˜ f , U 2 = S 2 ( s 1 ) P 2 2 + ˜ D P 1 + ˜ E P 2 + ˜ F , (1.11) where S i ( s i ) = 1 (Φ i q i ) 2 (( as i − A )(Φ i q 1 ) 2 + 2( bs i − B )Φ i q 1 Φ i q 2 + ( cs i − C )(Φ i q 2 ) 2 ) (1.12) W e alulate a P oisson bra k et b et w een H and K. Then the o eien t of P 2 1 , P 2 2 , P 1 P 2 equal to zero i ˜ d = 2 S 1 ( s 1 ) ∂ F ( s 1 , s 2 ) ∂ s 1 , ˜ e = p S 1 ( s 1 ) S 2 ( s 2 ) Z s 1 ( s 1 − s 2 ) , ˜ D = − p S 1 ( s 1 ) S 2 ( s 2 ) Z s 2 ( s 1 − s 2 ) , ˜ E = 2 S 2 ( s 2 ) ∂ F ( s 1 , s 2 ) ∂ s 2 where Z ( s 1 , s 2 ) , F ( s 1 , s 2 ) - some funtions. and Z s 1 ,s 2 = Z s 1 − Z s 2 2( s 2 − s 1 ) (1.13) W e apply the anonial transformation ˆ P 1 = P 1 + ∂ F ( s 1 , s 2 ) ∂ s 1 , ˆ P 2 = P 2 + ∂ F ( s 1 , s 2 ) ∂ s 2 to equate ˜ d, ˜ E to zero. Then the o eien t of P 1 , P 2 equal to zero i U 1 , U 2 ha v e the form as in form ulation of Theorem 1. And nally the free o eien t in P oisson bra k et equals to zero i the equation (1.7) of the Theorem 1 is fullled just as exp eted. The general analytial solution of Euler - Darb oux equation ( 1.6) has near the line of singularities x = y the follo wing expansion: Z ( x, y ) = A + ln ( x − y ) B , A = ∞ X 0 a i ( x + y ) ( x − y ) 2 i , B = ∞ X 0 b i ( x + y ) ( x − y ) 2 i , where a 0 and a 1 - some funtion. The other o eien ts an b e expressed b y these t w o funtions and their deriv ativ es. F or example, b 0 = 1 2 a ′′ 0 . W e insert this expan tion in to (1.7) to obtain B = 0 . It is easy to  he k that an y solution of the equation (1.6) with B = 0 has the form Z ( x, y ) = z 0 + δ ( x + y ) + ( x − y ) 2 ∞ X k =0 g (2 k ) ( x + y ) 2 (2 k ) k !( k + 1)! ( x − y ) 2 k , (1.14) where g ( x ) - some funtion and z 0 , δ - some onstan ts. W e all the funtion g ( x ) as gener ating funtion for (1.14). Without the loss of generalit y w e  ho ose z 0 = 0 . The parameter δ , is v ery imp ortan t for lassiation of hamiltonians from Theorem 1. 3 W e nd all the funtions Z , orresp onding the rational generating funtions g . Cho osing g ( x ) = x n , w e obtain the innite set of p olynomial solutions Z ( n ) for (1.6). In partiular g ( x ) = 1 ⇐ ⇒ Z (0) ( x, y ) = ( x − y ) 2 g ( x ) = x ⇐ ⇒ Z (1) ( x, y ) = ( x + y )( x − y ) 2 , g ( x ) = x 2 ⇐ ⇒ Z (2) ( x, y ) = 1 4  ( x − y ) 2 + 4( x + y ) 2  ( x − y ) 2 . All set an b e obtained b y using 'reating' op erator x 2 ∂ ∂ x + y 2 ∂ ∂ y − 1 2 ( x + y ) , ating on Z (0) . The rational funtions g ( x ) = ( x − µ ) − n reate one more lass of exat solution of equation (1.6). F or example g µ ( x ) = 1 4 1 x − 2 µ ⇐ ⇒ Z µ ( x, y ) = p ( µ − x )( µ − y ) + 1 2 ( x + y ) − µ. The solution orresp onding the p oles of order n ≥ 2 , an b e obtained b y dieren tiating the last form ula b y parameter µ. Beause funtion Z is linear b y g w e obtained the solution Z with rational generating funtion g ( x ) = P i c i x i + P i,j d ij ( x − µ i ) − j . Hyp othesis 1. F or all Hamiltonians (1.3)-(1.7 ) generating funtion g is rational and has the form g ( x ) = P ( x ) S ( x ) , where P è S - some p olynomials with deg P < 5 , deg S < 6 . In pap ers [5, 6 ℄ the follo wing solution of the system ( 1.6), (1.7) w as onsidered: Z ( x, y ) = x + y , S 1 ( x ) = S 2 ( x ) = 6 X i =0 c i x i , f 1 ( x ) = f 2 ( x ) = − 3 4 c 6 x 4 − 1 2 c 5 x 3 + 2 X i =0 k i x i , where c i , k i - some onstan ts. A v ery imp ortan t fat is that Clebs h top and so (4) -S hottky- Manak o v top [8, 9, 10 ℄ are the partiular ases of this mo del [6℄. In pap er [6℄ a full solution of Hamilton - Jaobi equation of this mo del w as obtained in the form of some kind of separation of v ariables on a non-h yp erellipti urv e of gen us 4. 2 Univ ersal solution of Hamilton-Jaobi equation Let H and K ha v e the form (1.3)-(1.5 ). Consider system H = e 1 , K = e 2 , where e i - some onstan ts. Let p 1 = F 1 ( x, y ) , p 2 = F 2 ( x, y ) - b e its solution. W e use short notation x è y orresp onding q 1 è q 2 . Jaobi's lemma giv es that if { H , K } = 0 , then ∂ F 1 ∂ y = ∂ F 2 ∂ x . T o nd an ation S ( x, y , e 1 , e 2 ) , it is enough to solv e the follo wing system ∂ ∂ x S = F 1 , ∂ ∂ y S = F 2 . 4 W e rewrite the system H = e 1 , K = e 2 in the form p 2 1 + ap 2 + b = 0 , p 2 2 + Ap 1 + B = 0 , (2.15) where a = Z x x − y s S 2 ( y ) S 1 ( x ) , A = − Z y x − y s S 1 ( x ) S 2 ( y ) b = − Z 2 x 4( x − y ) 2 + V 1 − e 1 x + e 2 S 1 ( x ) , B = − Z 2 y 4( x − y ) 2 + V 2 − e 1 y + e 2 S 2 ( y ) . It easy to nd that 2 b y + Aa x + 2 aA x = 0 , 2 Aa y + aA y + 2 B x = 0 . (2.16) Using (1.6 ) and (1.7 ), it is easy to obtain the follo wing iden tit y Ab x − aB y + 2 A x b − 2 a y B = 0 . (2.17) Using a standard te hnique of Lagrange resolv en ts (see f.e. [11℄), w e rewrite system (2.15 ) to a system uv = 1 4 aA, (2.18) Au 3 + 4 b a u 2 v − 4 B A uv 2 − av 3 = 0 , (2.19) that is equiv alen t to the qubi equation on u 2 . Let ( u k , v k ) , k = 1 , 2 , 3 b e the solutions of (2.18 ), (2.19 ) su h that u 2 1 + u 2 2 + u 2 3 = − b, v 2 1 + v 2 2 + v 2 3 = − B u 1 u 2 u 3 = − 1 8 a 2 A, v 1 v 2 v 3 = − 1 8 A 2 a. Then, form ulas p 1 = u 1 + u 2 + u 3 , p 2 = v 1 + v 2 + v 3 ; p 1 = u 3 − u 1 − u 2 , p 2 = v 3 − v 1 − v 2 ; p 1 = u 2 − u 1 − u 3 , p 2 = v 2 − v 1 − v 3 ; p 1 = u 1 − u 2 − u 3 , p 2 = v 1 − v 2 − v 3 dene four solutions of (2.15). Consider the rst of them. Lemma 1. F or i = 1 , 2 , 3 fol lowing e quations ar e ful lle d ∂ u i ∂ y = ∂ v i ∂ x . Pro v e. Dieren tiating equations (2.18 ) and (2.19 ) on x and y , w e nd u y and v x as the funtions on u and v . Then expressing v through u, w e obtain that u y = v x is equiv alen t to iden tities (2.16) and (2.17 ).  Lemma 1 means, that in v ariables u 1 , u 2 , u 3 w e nd partiular separation v ariables. Really S = S 1 + S 2 + S 3 , where S is the ation, and funtions S i dened from a system ∂ ∂ x S i = u i , ∂ ∂ y S i = v i . 5 Let's u = 1 2 Z x x − y s y − ξ x − ξ , v = − 1 2 Z y x − y s x − ξ y − ξ . It easy to see that pair ( u, v ) for all ξ are a solution of (2.18 ). If Z is a solution of (1.6), then ∂ u ∂ y = ∂ v ∂ x . Using this fat w e in tro due a funtion σ ( x, y , ξ ) so that ∂ σ ∂ x = u, ∂ σ ∂ y = v . In a ase of rational funtion g , orresp onding funtion Z is expressed through quadrati radials and the funtion σ an b e obtained. Let's Y = ∂ σ ∂ ξ . After m ultipliation of expression (2.19) b y expression − 2 p S 1 ( x ) p S 2 ( y ) √ x − ξ √ y − ξ ( x − y ) Z x Z y , left side of (2.19 ) an b e written in the form − e 2 + e 1 ξ + y − ξ x − y  V 1 − S 1 ( x ) Z 2 x 4( x − ξ )( x − y )  − x − ξ x − y  V 2 + S 2 ( y ) Z 2 y 4( y − ξ )( x − y )  . (2.20) Prop osition 1 . L et the expr ession (1.6 ), (1.7 ) b e full le d . Then the expr ession ( 2.20 ) is a funtion of Y and ξ v ariables only . Pro v e . W e assign the funtion (2.20) as Ψ( x, y , ξ ) . Consider Jaobian J = ∂ Ψ ∂ x ∂ Y ∂ y − ∂ Ψ ∂ y ∂ Y ∂ x . W e  hange ∂ Y ∂ y and ∂ Y ∂ x to ∂ v ∂ ξ and ∂ u ∂ ξ , resp etevily , then Jaobian J equals to zero iden tially taking in to aoun t (1.6 ), (1.7 ).  Due to Prop osition 1, the relation Ψ( x, y , ξ ) = 0 an b e rewritten in the form φ ( ξ , Y ) = 0 . One an nd the funtion φ b y assuming y = x . Equation φ ( ξ , Y ) = 0 denes a urv e, and the dieren tials of this urv e dene the funtion of ation S. W e note ξ k ( x, y ) , where k = 1 , 2 , 3 , the ro ots of ubi equation Ψ( x, y , ξ ) = 0 . Theorem 2. The funtion of ation S has the form S ( x, y ) = 3 X k =1  σ ( x, y , ξ k ) − ξ k Z Y ( ξ ) dξ  , (2.21) wher e Y ( ξ ) - alebr ai funtion on the urve φ ( ξ , Y ) = 0 . Pro v e. W e obtain ∂ ∂ x S ( x, y ) = 3 X k =1 σ x ( x, y , ξ k ) + 3 X k =1 { σ ξ ( x, y , ξ k ) − Y ( ξ k ) } ξ k ,x = 3 X k =1 u k = p 1 . 6 Analogously ∂ ∂ y S ( x, y ) = p 2 .  3 Case of ubis Consider a ase when the urv e(2.20 ) an b e written in the form ˜ φ ( ξ , η ) = 0 ⇔ φ ( ξ , Y ) , so that p oin ts ( ξ 1 , η 1 ) , ( ξ 2 , η 2 ) , ( ξ 3 , η 3 ) lie on a straigh t line, that equiv alen t to denition η = ξ a ( x, y ) + b ( x, y ) , its substitution in to ˜ φ, giv es a urv e Ψ( x, y , ξ ) = 0 . F orm ula (2.20) giv es a urv e in a new v ariables ξ , η − e 2 + e 1 ξ + C 2 ( ξ ,η ) C 1 ( ξ ,η ) = 0 , where C 1 ( ξ , η ) → 0 at x → 0 or y → 0 . Using rev ersible urv e equation C 1 ( ξ , η ) = 0 → η = f ( ξ ) using η , w e nd the expressions for a ( x, y ) , b ( x, y ) a ( x, y ) = f ( x ) − f ( y ) x − y , b ( x, y ) = y f ( x ) − xf ( y ) x − y On the other hand the equiv alene of the urv e φ ( ξ , Y ) = 0 è ˜ φ ( ξ , η ) = 0 giv es Y x η y = Y y η x ⇔ u ξ η y = v ξ η x ⇔ ( ξ − y ) Z x η y = ( x − ξ ) Z y η x , or Z = Z ( a ) , b x = − y a x , b y = − x a y . 4 Examples In this Setion w e onsider all the pairs of Hamiltonians kno wn at the momen t (1.3 )-(1.7 ). 4.1 Class 1 F or the mo dels of this lass S 1 = S 2 = S, f 1 = f 2 = f . (4.22) Theorem 3. L et g = ˜ G S , ˜ G = G − δ 10 S ′ , f = − 4 ˜ G 2 S − 4 δ 3 ˜ G ′ − δ 2 12 S ′′ , wher e S ( x ) = s 5 x 5 + s 4 x 4 + s 3 x 3 + s 2 x 2 + s 1 x + s 0 , G ( x ) = g 3 x 3 + g 2 x 2 + g 1 x + g 0 , wher e s i , g i , δ - some  onstants. Then funtions S, f and funtion Z ,  orr esp onding (see. 1) gener ation funtion g , full l the systems ( 1.6), (1.7). 7 Remark. P arameter δ from Theorem 3 oinsides with parameter δ from (1.14 ). Consider the ase δ = 0 in the form ula (1.14 ), Then all pairs of Hamiltonians (1.3)-(1.7 ), (4.22 ), that fulll this ondition are desrib ed b y Theorem 3. Consider a general ase S ( x ) = s 5 ( x − µ 1 )( x − µ 2 )( x − µ 3 )( x − µ 4 )( x − µ 5 ) , where s 5 6 = 0 and all ro ots µ i of p olynomial S are distint. then the funtion Z has the form Z ( x, y ) = 5 X i =1 ν i p ( µ i − x )( µ i − y ) , (4.23) where ν i - some onstan ts. Co etien ts g i and δ are expressed through onstan ts ν j from (1.14 ). F or example, 2 δ = − P ν i . F untion f is dened b y f ( x ) = − 1 16 5 X i =1 ν 2 i S ′ ( µ i ) x − µ i + k 1 x + k 0 , where k 1 , k 0 - some onstan ts. Calulation for a funtion (4.23 ) giv es σ ( x, y , ξ ) = − 1 2 5 X i =1 ν i log √ x − ξ √ y − µ i + √ y − ξ √ x − µ i √ x − y √ µ i − ξ , (4.24) Y = 1 4 N X i =1 ν i p ( x − µ i )( y − µ i ) ( ξ − µ i ) p ( x − ξ )( y − ξ ) . Algebrai urv e has the form of h yp erellipti urv e of gen us = 2 φ ( Y , ξ ) = S ( ξ ) Y 2 + f ( ξ ) − ξ e 1 + e 2 = 0 Steklo v top on so (4) [12℄ is a partiular ase of Theorem 3. 4.2 Class 2 F untions Z for the mo dels of this lass are the sp eial ases of the funtions Z of Class 1. But this Class on tains m u h more parameters them Theorem 3. Su h funtions Z an b e dened as the solutions of system Z xy = Z x − Z y 2( y − x ) = 1 3 U ( Z ) Z x Z y , (4.25) where U - some funtions of one v ariable. Remark. It easy to see that this lass of solutions of Euler - Darb oux equation Z xy = Z x − Z y 2( y − x ) oinide with the lass of solutions of the form Z = F  h ( x ) − h ( y ) x − y  , 8 where F and h - some funtions of one v ariable and U = F ′′ /F ′ 2 . Lemma. The system (4.25 ) is  omp atible if and only if U = 3 2 B ′ B , B ( Z ) = b 2 Z 2 + b 1 Z + b 0 , wher e b i - some  onstants . In a ase deg B = 2 Z ( x, y ) = p ( x − µ 1 )( y − µ 1 ) + p ( x − µ 2 )( y − µ 2 ) , (4.26) where b 2 = 1 , b 1 = 0 , b 0 = − ( µ 1 − µ 2 ) 2 . If deg B = 1 , then Z ( x, y ) = √ x y + 1 2 ( x + y ) , (4.27) b 1 = 1 , b 2 = b 0 = 0 . If deg B = 0 , then Z ( x, y ) = x + y . (4.28) 1. Consider funtion Z of the form (4.26). Then S ( x ) = ( x − µ 1 )( x − µ 2 ) P ( x ) + ( x − µ 1 ) 3 / 2 ( x − µ 2 ) 3 / 2 Q ( x ) , deg P ≤ 3 , deg Q ≤ 2 , è f ( x ) = f 0 + f 1 x + k 2 ( x − µ 1 ) 1 / 2 ( x − µ 2 ) 1 / 2 + ( µ 2 − µ 1 ) 16 n P ( µ 1 ) x − µ 1 − P ( µ 2 ) x − µ 2 o + ( µ 2 − µ 1 ) 32 ( x − µ 1 ) 1 / 2 ( x − µ 2 ) 1 / 2 n Q ( µ 1 ) x − µ 1 − Q ( µ 2 ) x − µ 2 o . In a ase when Q = 0 , k 2 = 0 , These form ulas oinside with orresp onding form ulas of Class 1. The funtions σ , Y are dened the same form ula (4.24) as for Class 1 : σ ( x, y , ξ ) = − 1 2 2 X i =1 log √ x − ξ √ y − µ i + √ y − ξ √ x − µ i √ x − y √ µ i − ξ , Y = 1 4 2 X i =1 p ( x − µ i )( y − µ i ) ( ξ − µ i ) p ( x − ξ )( y − ξ ) . Algebrai urv e in this ase has the form [ S R ( ξ ) + η S I ( ξ )] Y 2 − [ k R ( ξ ) + η k I ( ξ )] = 0 , (4.29) where S R ( x ) = ( x − µ 1 )( x − µ 2 ) P ( x ) , S I ( x ) = ( x − µ 1 )( x − µ 2 ) Q ( x ) , k R ( x ) = − e 2 + e 1 x − f 0 − f 1 x − ( µ 2 − µ 1 ) 16 n P ( µ 1 ) x − µ 1 − P ( µ 2 ) x − µ 2 o , k I ( x ) = k 2 − 1 32 ( µ 1 − µ 2 ) 2 − 1 16 ( µ 1 − µ 2 ) n Q ( µ 1 ) x − µ 1 − Q ( µ 2 ) x − µ 2 o , 1 η = 1 √ ξ − µ 1 √ ξ − µ 2 s 1 − ( µ 1 − µ 2 ) 2 16( ξ − µ 1 ) 2 ( ξ − µ 2 ) 2 Y 2 . 9 Expressing Y as a funtion of ( ξ , η ) and substituting to (4.29 ), w e obtain 10-parameter ubi in ( ξ , η ) , v ariables. So in a general ase the urv e φ ( Y , ξ ) = 0 , is a o v ering o v er an ellipti urv e W e obtain η = ξ − µ 1 √ x − µ 1 √ x − µ 2 + √ y − µ 1 √ y − µ 2 + ξ − µ 2 √ x − µ 2 √ x − µ 1 + √ y − µ 2 √ y − µ 1 , therefore p oin ts ( ξ 1 , η 1 ) , ( ξ 2 , η 2 ) , ( ξ 3 , η 3 ) lie on a straigh t line. 2. F or the funtion Z of the form (4.27 ) w e ha v e S ( x ) = xP ( x ) + x 3 / 2 Q ( x ) , deg P ≤ 3 , deg Q ≤ 2 , f ( x ) = − 1 16 x P ( x ) − 1 32 √ x Q ( x ) + f 1 x + f q √ x + f 0 . The funtion Y is dened b y Y = ξ + √ x √ y 4 ξ √ x − ξ √ y − ξ . The urv e in this ase an b e written in the form (4.29 ), where S R ( x ) = xP ( x ) , S I ( x ) = xQ ( x ) , k R ( x ) = − e 2 + e 1 x − f 0 − f 1 x + 1 16 x P ( x ) , k I ( x ) = 1 16 x Q ( x ) − f q , η = 4 Y ξ 3 / 2 p 16 Y 2 ξ 2 − 1 . In ( ξ , η ) v ariables it also has the form of arbitrary ubi. F orm ula η = ξ + √ xy √ x + √ y giv es the fat that p oin ts ( ξ 1 , η 1 ) , ( ξ 2 , η 2 ) , ( ξ 3 , η 3 ) lie on a straigh t line 3. F or the funtion Z , giv en b y (4.28 ), w e obtain S ( x ) = s 6 x 6 + s 5 x 5 + s 4 x 4 + s 3 x 3 + s 2 x 2 + s 1 x + s 0 , f ( x ) = − 1 40 S ′′ ( x ) − 1 32 √ x Q ( x ) + f 2 x 2 + f 1 x + f 0 . In this ase Y = 1 2 √ x − ξ √ y − ξ . Algebrai urv e S ( ξ ) Y 6 − F ( ξ ) Y 4 −  1 8 F ′′ ( ξ ) + 7 1920 S I V ( ξ ) − k 2 2  Y 2 − s 6 64 = 0 , F ( ξ ) = − e 2 + e 1 ξ − f ( ξ ) and in ( ξ , η ) , v ariables where η = ξ 2 − 1 4 Y 2 , has the form of arbitrary ubi. Beause η = ξ ( x + y ) − xy , p oin ts ( ξ 1 , η 1 ) , ( ξ 2 , η 2 ) , ( ξ 3 , η 3 ) b elong to a straigh t line. 10 4.3 Class 3 W e in tro due 'non-symmetrial Hamiltonian ( 1.3)-(1.7 ) if S 1 ( x ) 6 = S 2 ( x ) , or f 1 ( x ) 6 = f 2 ( x ) . Theorem 4. [7℄ In non-symmetri al  ase the funtions Z , S i , f i is the solutions of (1.6 ), (1.7 ) if and only if δ = 0 , g = 1 H , S 1 , 2 = W H ± M H 3 / 2 , f 1 , 2 = − 4 W H ∓ 2 M H − 1 / 2 ± aH 1 / 2 , wher e g - gener ation funtion of Z , W ( x ) = w 3 x 3 + w 2 x 2 + w 1 x + w 0 , H ( x ) = h 2 x 2 + h 1 x + h 0 , M ( x ) = m 2 x 2 + m 1 x + m 0 . Her e w i , h i , m i , a - some  onstants . Consider the general ase H ( x ) = ( x − µ 1 )( x − µ 2 ) . Algebrai urv e in this ase is dened b y Ψ( ξ , Y ) = − e 2 + e 1 ξ − R W ( ξ ) 2( ξ − µ 1 )( ξ − µ 2 )( µ 2 − µ 1 ) + 4 M ( ξ ) √ 2 Y √ ξ − µ 1 √ ξ − µ 2 ( µ 2 − µ 1 ) 3 / 2 √ R + 8 b √ 2 Y ( ξ − µ 1 ) 3 / 2 ( ξ − µ 2 ) 3 / 2 √ R √ µ 2 − µ 1 = 0 , (4.30) where Y = p ( x − µ 1 )( y − µ 1 ) ( ξ − µ 1 ) p ( x − ξ )( y − ξ ) − p ( x − µ 2 )( y − µ 2 ) ( ξ − µ 2 ) p ( x − ξ )( y − ξ ) , R = 1 6 ( ξ − µ 1 ) 2 ( ξ − µ 2 ) 2 Y 2 − ( µ 1 − µ 2 ) 2 . Substituting Y = 1 4 ( µ 1 − µ 2 ) 3 2 η ( ξ − µ 2 )( ξ − µ 1 ) p η 2 ( µ 2 − µ 1 ) − 8( ξ − µ 1 )( ξ − µ 2 ) in to (4.30 ), W e obtain the ubi in v ariables ( ξ , η ) with a full set of ten indep enden t parameters. It easy to see that η = a ( x, y ) ξ + b ( x, y ) , where a, b - some funtions. Therefore in the ases of Class 2 and 3 the algebrai urv e is non-h yp erellipti o v ering o v er the ellipti urv e. The dynamis of the p oin ts ( ξ 1 , Y 1 ) , ( ξ 2 , Y 2 ) , ( ξ 3 , Y 3 ) on this urv e (see. theorem 2) denes the follo wing ondition: their pro jetion on the ellipti base ( ξ 1 , η 1 ) , ( ξ 2 , η 2 ) , ( ξ 3 , η 3 ) lies on the straigh t line. Hyp othesis 2. An y pair of the Hamiltonians (1.3)-(1.7 ) b elongs to one of ab o v e three lasses. 5 App endix 1. Steklo v top W e sho w that the ase of Steklo v top on so (4) is a partiular ase of Class 1 after restrition on the sympleti leafs. Hamiltonian and the additional in tegral in this ase ha v e the form H = ( ~ S 1 , A ~ S 1 ) + ( ~ S 1 , B ~ S 2 ) , K = ( ~ S 1 , ¯ A ~ S 1 ) + ( ~ S 1 , ¯ B ~ S 2 ) , 11 where A = − α 2 diag ( 1 α 2 1 , 1 α 2 2 , 1 α 2 3 ) , B = α diag ( α 1 , α 2 , α 3 ) , ¯ A = − diag ( α 2 1 , α 2 2 , α 2 3 ) , ¯ B = α d iag ( 1 α 1 , 1 α 2 , 1 α 3 ) , α = α 1 α 2 α 3 . Here ~ S i - three-dimensional v etors with omp onen ts S α i . It is easy to see that, H and K omm ute under a spin P oisson bra k et { S α i , S β j } = κ ε αβ γ δ ij S γ i . It is on v enien t to  hose κ = − 2 i. W e x the Casimirs for spin bra k et: ( ~ S k , ~ S k ) = j 2 k , k = 1 , 2 . Then the form ulas ~ S k = π k ~ K ( Q k ) + j k 2 ~ K ′ ( Q k ) , ã äå ~ K ( Q ) = (( Q 2 − 1) , i ( Q 2 + 1) , 2 Q ) , dene the Darb oux o ordinate π 1 , π 2 , Q 1 , Q 2 for the simpleti leaf of P oisson manifold with o ordinates ~ S k , k = 1 , 2 . As this transformation is linear b y momen ta π k , as a result, w e obtain a pair of omm uting Hamiltonians quadrati in momen ta under the bra k et { π α , Q β } = δ αβ . Consider the anonial transformation that transforms their pair to the form (1.3)-(1.5 ), (4.22 ). W e apply the anonial transformation P 1 = π 1 p r ( Q 1 ) , P 2 = π 2 p R ( Q 2 ) , dX = dQ 1 p r ( Q 1 ) , d Y = dQ 2 p R ( Q 2 ) , where r ( Q 1 ) = ( ~ K ( Q 1 ) , A ~ K ( Q 1 )) , R ( Q 2 ) = ( ~ K ( Q 2 ) , ¯ A ~ K ( Q 2 )) , to obtain H = P 2 1 + 2 P 1 P 2 V + j 2 P 1 V Y + j 1 P 2 V X + 1 2 j 1 j 2 V X,Y + j 2 1 6  g ′′ 1 ( X ) g 1 ( X ) − 3 2 ( g ′ 1 ( X ) g 1 ( X ) ) 2  , K = P 2 2 + 2 P 1 P 2 W + j 2 P 1 W Y + j 1 P 2 W X + 1 2 j 1 j 2 W X,Y + j 2 2 6  g ′′ 2 ( Y ) g 2 ( Y ) − 3 2 ( g ′ 2 ( Y ) g 2 ( Y ) ) 2  . where V ( X , Y ) = ( ~ K ( Q 1 ) , B ~ K ( Q 2 )) p r ( q 1 ) p R ( Q 2 ) , W ( X , Y ) = ( ~ K ( Q 1 ) , ¯ B ~ K ( Q 2 )) p r ( Q 1 ) p R ( Q 2 ) , g 1 ( X ) = p r ( Q 1 ) , g 2 ( Y ) = p R ( Q 2 ) . W e apply the anonial transformation ( P 1 , P 2 , X , Y ) → ( p 1 , p 2 , x, y ) of the form dX = 1 2  dx p S ( x ) + dy p y S ( y )  , d Y = − 1 2  dx p xS ( x ) + dy p S ( y )  , P 1 = 2 √ x − √ y h p 1 − j 1 + j 2 4( x − y ) r y x  p S ( x ) −  p 2 + j 1 + j 2 4( x − y ) r x y  p S ( y ) i , 12 P 2 = 2 √ x − √ y h p 1 − j 1 + j 2 4( x − y ) r y x  p y S ( x ) −  p 2 + j 1 + j 2 4( x − y ) r x y  p x S ( y ) i , to obtain (1.3)-(1.5 ), (4.22), where S ( x ) = − 4 x (1 + α 2 1 x )(1 + α 2 2 x )(1 + α 2 3 x ) , Z ( x, y ) = − 1 2 j 1 ( x + y ) − j 2 √ xy , f ( x ) = 1 4  j 2 1 α 2 x 2 + j 2 2 x  − j 2 2 1 4 α 2  1 α 2 1 + 1 α 2 2 + 1 α 2 3  x. A  kno wledgemen ts. The author are grateful to E V F erap on to v for useful disussion. The resear h w as partially supp orted b y RFBR gran t 05-01-00189 and NSh 6358.2006.2. Ñïèñîê ëèòåðàòóðû [1℄ B. Dorizzi, B. Grammatios, A. Ramani and P . 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