Separation of variables in the generalized 4th Appelrot class

Separation of v ariables in the generalized 4th App elrot class ∗ Mikhail P . Kharl a mov Gagarin Str e et 8, V olgo gr ad, 40013 1 Russia e-mail: mharla mov@vags.ru 28.0 1.07 Abstract W e consider the analogue of the 4th App elrot class of mot ions of the Ko w alevski top for the c ase of t wo constan t force fields. The tr a jectories of this family fill the four-dimensional surface O in the six-dimensional p hase sp ace. The constan ts of thr ee first integ rals in in- v olution restricted to this surface fi ll one of the sheets of the bifu rcation d iagram in R 3 . W e p oint out the p air of p artial integ rals t o obtain the explicit parametric equations of this sheet. Th e ind uced system on O is s ho wn to b e Hamiltonian with tw o degrees of fr eedom ha ving the thin set of p oin ts where the induced symplectic structure degenerates. The region of existence of motions in terms of the int egral constan ts is found . W e pro vide the separation of v ariables on O and the alge braic form ulae for the initial p hase v ariables. Key w ords and phras es: Kowalevski top, double field, Appelr ot classes, se pa ration o f v a riables MSC2000 n um bers : 7 0 E17, 70 G4 0 DOI: 10.1134 /S1560 354707030021 1 Preliminaries The equations of motion of the Ko w alevski top in t w o constant fields, expressed in the reference system O e 1 e 2 e 3 of the principal axes of inertia at the fixed p oint O , 2 ˙ ω 1 = ω 2 ω 3 + β 3 , 2 ˙ ω 2 = − ω 1 ω 3 − α 3 , ˙ ω 3 = α 2 − β 1 , ˙ α 1 = α 2 ω 3 − α 3 ω 2 , ˙ β 1 = β 2 ω 3 − β 3 ω 2 , ˙ α 2 = α 3 ω 1 − α 1 ω 3 , ˙ β 2 = β 3 ω 1 − β 1 ω 3 , ˙ α 3 = α 1 ω 2 − α 2 ω 1 , ˙ β 3 = β 1 ω 2 − β 2 ω 1 (1.1) can without loss of generality b e restricted to the phase space P 6 ⊂ R 9 ( ω , α , β ) defined b y the geometric in tegrals α 2 1 + α 2 2 + α 2 3 = a 2 , β 2 1 + β 2 2 + β 2 3 = b 2 , α 1 β 1 + α 2 β 2 + α 3 β 3 = 0 (1.2) ∗ Regular and Cha otic Dyna mics , 20 07, V ol. 1 2, No. 3, pp. 26 7-280 1 (see [1] for details). This sy stem is completely integrable due to the existence of three in tegrals in in v olution [2, 3] H = ω 2 1 + ω 2 2 + 1 2 ω 2 3 − ( α 1 + β 2 ) , K = ( ω 2 1 − ω 2 2 + α 1 − β 2 ) 2 + (2 ω 1 ω 2 + α 2 + β 1 ) 2 , G = 1 4 ( M 2 α + M 2 β ) + 1 2 ω 3 M γ − b 2 α 1 − a 2 β 2 , (1.3) where M α = 2 ω 1 α 1 + 2 ω 2 α 2 + ω 3 α 3 , M β = 2 ω 1 β 1 + 2 ω 2 β 2 + ω 3 β 3 , M γ = 2 ω 1 ( α 2 β 3 − α 3 β 2 ) + 2 ω 2 ( α 3 β 1 − α 1 β 3 ) + ω 3 ( α 1 β 2 − α 2 β 1 ) . (1.4) F or the general case a > b > 0 (1.5) the explicit in tegration has not b een found y et. It is natural to study first the in v a rian t subman- ifolds in P 6 suc h that the induced system has only t w o degrees of f reedom. It is pro v ed in [4, 1] that there exist only three submanifolds M , N , O of this type. The union M ∪ N ∪ O coincides with the set of critical p oin ts of the in tegra l map H × K × G : P 6 → R 3 . (1.6) If b = 0 ( the classical Kow alevski case [5]), then the critical set of the map (1.6) consists of the motions that b elong to the so-called fo ur App elrot classes [6]. The set M , first found in [2] as the zero lev el o f the in tegral K , generalizes the 1st App elrot class. T he phase t o p ology o f the system induced on M w as studied in [7]. The dynamical system o n N generalizing t he 2nd a nd 3rd App elrot classes was explicitly in tegrated in [8]. The prese n t w ork conside rs the restriction of the system (1.1) to the in v ariant subse t O . In tro duce the complex phase v ariables [9] ( i 2 = − 1): x 1 = ( α 1 − β 2 ) + i ( α 2 + β 1 ) , x 2 = ( α 1 − β 2 ) − i ( α 2 + β 1 ) , y 1 = ( α 1 + β 2 ) + i ( α 2 − β 1 ) , y 2 = ( α 1 + β 2 ) − i ( α 2 − β 1 ) , z 1 = α 3 + iβ 3 , z 2 = α 3 − iβ 3 , w 1 = ω 1 + iω 2 , w 2 = ω 1 − iω 2 , w 3 = ω 3 . (1.7) The system (1.1) tak es the form x ′ 1 = − x 1 w 3 + z 1 w 1 , x ′ 2 = x 2 w 3 − z 2 w 2 , y ′ 1 = − y 1 w 3 + z 2 w 1 , y ′ 2 = y 2 w 3 − z 1 w 2 , 2 z ′ 1 = x 1 w 2 − y 2 w 1 , 2 z ′ 2 = − x 2 w 1 + y 1 w 2 , 2 w ′ 1 = − ( w 1 w 3 + z 1 ) , 2 w ′ 2 = w 2 w 3 + z 2 , 2 w ′ 3 = y 2 − y 1 . (1.8) Here the prime stands for d/d ( it ). The set O , b y the definition giv en in the w ork [1], includes as a prop er subset the followin g p oin ts w 1 = w 2 = 0 , z 1 = z 2 = 0 . (1.9) The in v ar ia n t relations (1.9) lead to the family of p endulum motions first found in [4] α = a ( e 1 cos θ − e 2 sin θ ) , β = ± b ( e 1 sin θ + e 2 cos θ ) , α × β ≡ ± ab e 3 , ω = dθ dt e 3 , d 2 θ dt 2 = − ( a ± b ) sin θ . (1.10) 2 The corresp onding v alues of the in tegrals (1.3) satisfy one of the follow ing g = ± ab h, k = ( a ∓ b ) 2 , h > − ( a ± b ) . (1.11) In the sequel w e use functions and expressions hav ing singularities at the p oints (1.9). There- fore, b y default, we ex clude the tra jectories (1.10). The remaining part of O can b e describ ed b y the pair of equations R 1 = 0 , R 2 = 0 , (1.12) where R 1 = w 2 x 1 + w 1 y 2 + w 3 z 1 w 1 − w 1 x 2 + w 2 y 1 + w 3 z 2 w 2 , R 2 = ( w 2 z 1 + w 1 z 2 ) w 2 3 + h w 2 z 2 1 w 1 + w 1 z 2 2 w 2 + w 1 w 2 ( y 1 + y 2 ) + x 1 w 2 2 + x 2 w 2 1 i w 3 + + w 2 2 x 1 z 1 w 1 + w 2 1 x 2 z 2 w 2 + x 1 z 2 w 2 + x 2 z 1 w 1 + ( w 1 z 2 − w 2 z 1 )( y 1 − y 2 ) . (1.13) F or the deriv atives w e hav e, in virtue of (1.8), R ′ 1 = κ 2 R 2 , R ′ 2 = κ 1 R 1 , κ 1 = 1 2 w 1 w 2  ( w 1 w 2 w 3 + z 2 w 1 + z 1 w 2 ) 2 + w 1 w 2 ( x 2 w 2 1 + x 1 w 2 2 )  , κ 2 = 1 2 w 1 w 2 . (1.14) The equations (1 .14) straightforw ardly prov e that the set ( 1.12) is preserv ed b y the phase flo w (1 .1). Besides, it follow s that the P oisson brac k et { R 1 , R 2 } is a partial in tegral on O . The subset in O defined b y the equation { R 1 , R 2 } = 0 (1.15) is the set o f p oints at whic h the induced symplectic structure is degenerate. Belo w we calculate { R 1 , R 2 } using t he appropriate inte grals on O and sho w that the set (1.15) is of measure zero. Th us, the dynamical system induced on O by the sys tem ( 1.1) is almost ev erywhere a Hamiltonian system with t w o degrees of freedom. In particular, a lmost all its in t egra l manifolds consist of t w o- dimensional Liouville tori. 2 P artial integrals Recall that in the classical Kow alevski case ( β = 0) there exists the momen tum in tegral L = 1 2 I ω · α ( I = diag { 2 , 2 , 1 } ) . (2.1) Then the integral G b ecomes equal to L 2 . Let ℓ denote the constan t o f the in tegral (2.1). In App elrot’s notation the 4th class of esp e cia l ly r emarka b le motions is defined b y the follow ing conditions. (i) The second p olynomial of Kow alevski has a multiple ro o t . One of the Ko w alevsk i v ariables remains constan t and equal to the mu ltiple ro ot s of the corresp onding Euler resolve n t ϕ ( s ) = s ( s − h ) 2 + ( a 2 − k ) s − 2 ℓ 2 : ϕ ( s ) = 0 , ϕ ′ ( s ) = 0 . (2.2) 3 (ii) Tw o equatorial comp onents of the angular v elo cit y are constant: ω 1 ≡ − ℓ/s , ω 2 ≡ 0. Giv en β = 0, this fact can b e written in the form I ω · α I ω · e 1 = − s, I ω · β = 0 , I ω · e 2 = 0 . (2.3) The next statemen t establishes the conditions similar to ( 2.3) for the generalized top. Theorem 1. On e ach tr aje ctory b elo n ging to O the r atios I ω · α I ω · e 1 , I ω · β I ω · e 2 ar e c onstant and e qual to e ac h other. Pr o of. Let M = I ω , M j = I ω · e j ( j = 1 , 2 , 3) . (2.4) Due to (1.4) w e also hav e M α = I ω · α , M β = I ω · β . Then the first equation (1.12) yields M α M 1 − M β M 2 = 0 . (2.5) In tro duce the function S = − M α M 1 + M β M 2 M 2 1 + M 2 2 and calculate its time deriv ativ e: dS dt = − ( M 2 1 + M 2 2 ) ω 3 + 4 α 3 M 1 + 4 β 3 M 2 2( M 2 1 + M 2 2 ) 2 ( M α M 2 − M β M 1 ) . In virtue of (2.5) the right hand side is iden tically zero. T herefore S is a partia l integral o n O . Denote b y s the correspo nding constan t: M α M 1 + M β M 2 M 2 1 + M 2 2 = − s. (2.6) F rom (2.5), (2.6) w e obtain M α = − sM 1 , M β = − sM 2 (2.7) with the constan t v alue s . Note that according to (2.7) the function S can b e written in either of the represen t ations S = − 1 4  M α + iM β ω 1 + iω 2 + M α − iM β ω 1 − iω 2  = − 1 2 M α + iM β ω 1 + iω 2 = − 1 2 M α − iM β ω 1 − iω 2 . (2.8) Theorem 2. On the set O the system (1.1) has the p artial inte gr al T = 1 2 ( M α M 1 + M β M 2 ) − 2( α 1 β 2 − α 2 β 1 ) + a 2 + b 2 . (2.9) Pr o of. The time deriv ative of T d T dt = 1 4 ω 3 ( M α M 2 − M β M 1 ) v anishes on O due to (2.5). 4 Denote b y τ the constant of the in tegral T. Remark 1. In the work [1] e q uations similar to (1.12) wer e d e rive d fr om the c ond i tion that the function 2 G + ( τ − a 2 − b 2 ) H + s K (2.10) with L agr an ge’s m ultipliers s, τ has a critic al p oint on P 6 . Using the c o or dinates (1.7) we have fr om (2.8) , (2.9) S = − w 1 ( x 2 w 1 + y 1 w 2 + z 2 w 3 ) + w 2 ( y 2 w 1 + x 1 w 2 + z 1 w 3 ) 4 w 1 w 2 = = − x 2 w 1 + y 1 w 2 + z 2 w 3 2 w 2 = − y 2 w 1 + x 1 w 2 + z 1 w 3 2 w 1 , (2.11) T = 1 2 [ w 1 ( x 2 w 1 + y 1 w 2 + z 2 w 3 ) + w 2 ( y 2 w 1 + x 1 w 2 + z 1 w 3 )] + x 1 x 2 + z 1 z 2 = = − 2 S w 1 w 2 + x 1 x 2 + z 1 z 2 . (2.12) Comp aring (2.11) , (2.12) with the e xpr essions for s, τ given in [1] we se e that on O L agr ange’s multipliers in (2 .10) c oincide with the c onstants of the inte gr als S, T intr o duc e d he r e. Supp osing (1.5), introduce parameters p, r ( p > r > 0 ) suc h tha t p 2 = a 2 + b 2 , r 2 = a 2 − b 2 . (2.13) Let h, k , g denote the constants of the ge neral in tegrals (1.3). Then according to Remark 1, from the results of the w ork [1] w e o btain that on O h = p 2 − τ 2 s + s, k = τ 2 − 2 p 2 τ + r 4 4 s 2 + τ , g = p 4 − r 4 4 s + 1 2 ( p 2 − τ ) s. (2.14) These equations can be considered as para metric equations of the cor r esp onding bifurcation sheet of the in tegral map (1.6). Eliminating τ we h a v e ψ ( s ) = 0 , ψ ′ ( s ) = 0 , (2.15) where ψ ( s ) = s 2 ( s − h ) 2 + ( p 2 − k ) s 2 − 2 g s + p 4 − r 4 4 . If β = 0 ( p 2 = r 2 = a 2 ), then ψ ( s ) = s ϕ ( s ) and the conditions (2 .15) turn to (2.2). Therefore, the set of tra j ectories b elonging to O is the generalization of the set of the esp e cial ly r ema rkable motions of the 4th App elrot class. 3 P arametric equ ations of in tegral manifolds Due to the equations (2.14) the functions S, T form a complete system of first in tegrals on O . In particular, the equations of the in tegral manifold { ζ ∈ P 6 : H ( ζ ) = h, K ( ζ ) = k , G ( ζ ) = g } in this class of motions are equiv alent to the relations (1.12) and the equations S = s, T = τ . (3.1) 5 Using (2.5), (2.11) w e replace the equations R 1 = 0 , S = s b y ( y 2 + 2 s ) w 1 + x 1 w 2 + z 1 w 3 = 0 , x 2 w 1 + ( y 1 + 2 s ) w 2 + z 2 w 3 = 0 . (3.2) A t the same time, from (1.13) and (2.12) the system R 2 = 0 , T = τ is equiv alen t to x 2 z 1 w 1 + x 1 z 2 w 2 + ( τ − x 1 x 2 ) w 3 = 0 , (3.3) 2 s w 1 w 2 − ( x 1 x 2 + z 1 z 2 ) + τ = 0 . (3.4) T o obtain a closed sy stem of equations we m ust add the g eometric integrals (1.2). In terms of the v ariables (1.7) w e get z 2 1 + x 1 y 2 = r 2 , z 2 2 + x 2 y 1 = r 2 , (3.5) x 1 x 2 + y 1 y 2 + 2 z 1 z 2 = 2 p 2 . (3.6) The v ariables (1.7) b y definition satisfy x 2 = x 1 , y 2 = y 1 , z 2 = z 1 , w 2 = w 1 , w 3 ∈ R , (3.7) th us forming a space of real dimension 9. Sev en relations (3.2) – (3.6 ) define the inte gral manifold. In the case of indep endency of the in tegrals S, T this manifold is t wo-dimens ional and consists of Liouville tori filled with quasi-p erio dic motions. Let x = p x 1 x 2 , z = p z 1 z 2 , ξ = 2 s w 1 w 2 . (3.8) Then from (3.4) ξ = x 2 + z 2 − τ . (3.9) F rom (3.5), (3.8) w e obtain ( r 2 − x 1 y 2 )( r 2 − x 2 y 1 ) = z 4 , (3.10) and the equation (3.6) yields y 1 y 2 = 2 p 2 − x 2 − 2 z 2 . (3.11) Hence r 2 ( x 1 y 2 + x 2 y 1 ) = r 4 + 2 p 2 x 2 − ( x 2 + z 2 ) 2 . (3.12) Rewrite (3.5) in the form ( z 1 + z 2 ) 2 = 2 r 2 − ( x 1 y 2 + x 2 y 1 ) + 2 z 2 , ( z 1 − z 2 ) 2 = 2 r 2 − ( x 1 y 2 + x 2 y 1 ) − 2 z 2 (3.13) and substitute (3.12) to obtain r 2 ( z 1 + z 2 ) 2 = Φ 1 , r 2 ( z 1 − z 2 ) 2 = Φ 2 , (3.14) where Φ 1 = ( x 2 + z 2 + r 2 ) 2 − 2( p 2 + r 2 ) x 2 = ( ξ + τ + r 2 ) 2 − 2( p 2 + r 2 ) x 2 , Φ 2 = ( x 2 + z 2 − r 2 ) 2 − 2( p 2 − r 2 ) x 2 = ( ξ + τ − r 2 ) 2 − 2( p 2 − r 2 ) x 2 . (3.15) Note tha t the equilibria ω ≡ 0 of the system (1.1) are included in the family of motions (1.10). On the rest of the tra jectories in O t he determinan t o f the equations (3 .2), (3.3) in w j 6 ( j = 1 , 2 , 3 ) v a nishes iden t ically . Calculate this determinant and eliminate z 2 1 , z 2 2 from (3.5) a nd the pro duct y 1 y 2 from (3.11) to obtain 2 s [( r 2 x 1 − τ y 1 ) + ( r 2 x 2 − τ y 2 )] = − r 2 ( x 1 y 2 + x 2 y 1 )+ +2[2 s 2 ( τ − x 2 ) + p 2 ( τ + x 2 ) − τ ( x 2 + z 2 )] . (3.16) On the other hand, the direct calculation in view of (3.8), (3.11) giv es ( r 2 x 1 − τ y 1 )( r 2 x 2 − τ y 2 ) = r 4 x 2 + τ (2 p 2 − x 2 − 2 z 2 ) − r 2 τ ( x 1 y 2 + x 2 y 1 ) . (3.17) Denote σ = τ 2 − 2 p 2 τ + r 4 , χ = √ k > 0 . (3.18) F rom the second relation (2.14) w e ha ve the iden tity 4 s 2 χ 2 = σ + 4 s 2 τ . (3.19) In tro duce the complex conjugate v ariables µ 1 = r 2 x 1 − τ y 1 , µ 2 = r 2 x 2 − τ y 2 . (3.20) Eliminating x 1 y 2 + x 2 y 1 from (3.16), (3.17) with the help of (3.12) w e come to the syste m 2 s ( µ 1 + µ 2 ) = ξ 2 − 4 s 2 ( x 2 − τ ) − σ , µ 1 µ 2 = τ ξ 2 + σ x 2 − τ σ . (3.21) Cho ose µ ∗ 1 = p 2 sµ 1 − 4 s 2 τ , µ ∗ 2 = p 2 sµ 2 − 4 s 2 τ (3.22) to b e complex conjugate. Then the system (3.21) tak es the form ( µ ∗ 1 + µ ∗ 2 ) 2 = Ψ 1 , ( µ ∗ 1 − µ ∗ 2 ) 2 = Ψ 2 , (3.23) where Ψ 1 = ξ 2 − 4 s 2 ( x + χ ) 2 , Ψ 2 = ξ 2 − 4 s 2 ( x − χ ) 2 . (3.24) As w as to b e exp ected from dimensional reasoning, all phase v ariables could b e expressed in terms of t w o almost ev erywhe re indep enden t auxiliary v aria bles. The a b o v e form ulae emphasize the sp ecial role of the pair ( x, ξ ). F rom (3.22), (3.23) w e find µ 1 = 2 sτ + 1 8 s ( p Ψ 1 + p Ψ 2 ) 2 , µ 2 = 2 sτ + 1 8 s ( p Ψ 1 − p Ψ 2 ) 2 . (3.25) F urther on the calculation sequenc e is as follo ws. F rom (3.14) w e find z 1 , z 2 . Multiplying the equations (3.20) b y x 2 , x 1 resp ectiv ely and using (3.5) w e get x 2 µ 1 = r 2 x 2 − τ x 2 y 1 = r 2 ( x 2 − τ ) + τ z 2 2 , x 1 µ 2 = r 2 x 2 − τ x 1 y 2 = r 2 ( x 2 − τ ) + τ z 2 1 , 7 whence x 1 , x 2 are found. Substituting these v alues bac k to (3 .2 0) we find y 1 , y 2 . As a result, after some ob vious transformations w e obtain the follo wing expressions f o r the configuration v ariables: x 1 = 2 s r 2 4 r 4 ( x 2 − τ ) + τ ( p Φ 1 + p Φ 2 ) 2 16 s 2 τ + ( p Ψ 1 − p Ψ 2 ) 2 , x 2 = 2 s r 2 4 r 4 ( x 2 − τ ) + τ ( p Φ 1 − p Φ 2 ) 2 16 s 2 τ + ( p Ψ 1 + p Ψ 2 ) 2 , (3.26) y 1 = 2 s 4[2 τ ξ − τ ( x 2 − τ ) + σ ] − ( p Φ 1 − p Φ 2 ) 2 16 s 2 τ + ( p Ψ 1 − p Ψ 2 ) 2 , y 2 = 2 s 4[2 τ ξ − τ ( x 2 − τ ) + σ ] − ( p Φ 1 + p Φ 2 ) 2 16 s 2 τ + ( p Ψ 1 + p Ψ 2 ) 2 , (3.27) z 1 = 1 2 r ( p Φ 1 + p Φ 2 ) , z 2 = 1 2 r ( p Φ 1 − p Φ 2 ) . (3.28) Note that all radicals are algebraic. The fo rmal c hoice of signs in (3.26)–(3.28) is defined b y the initia l c hoice in the express ions for µ 1 , µ 2 , z 1 , z 2 . The p o lynomials Φ j , Ψ j ( j = 1 , 2) obv iously split to m ultipliers linear with res p ect to x, ξ . Then, typically , the pro jection of an in tegral manifold on to the ( x, ξ ) -plane has the form of a quadrangle. Fix any inner p oint ( x, ξ ) of suc h pro jection. Then the expre ssions (3.26)–(3.2 8) define eigh t p oin ts of the configuration space w ith differen t set of signs of the radicals p Φ 1 , p Φ 2 , p Ψ 1 Ψ 2 . T o find w 3 , use the energy in tegral. On account of (2.14) it t ak es the form 2 s w 2 3 + 4 s w 1 w 2 − 2 s ( y 1 + y 2 ) = 4 s 2 + 2 p 2 − 2 τ . Substitute 2 s w 1 w 2 from (3.4) and replace 2 p 2 b y its expre ssion fro m ( 3 .6) to obtain 2 s w 2 3 = D , (3.29) where D = ( y 1 + 2 s )( y 2 + 2 s ) − x 1 x 2 (3.30) is the determinant of ( 3 .2) with resp ect to w 1 , w 2 . In particular, this determinan t v anishes a long with w 3 . Due to this fact w e can express w 1 , w 2 either a s linear functions of w 3 , or in in v erse prop ortion to w 3 : w 1 = x 1 z 2 − ( y 1 + 2 s ) z 1 ( y 1 + 2 s )( y 2 + 2 s ) − x 2 w 3 = x 1 z 2 − ( y 1 + 2 s ) z 1 2 s w 3 , w 2 = x 2 z 1 − ( y 2 + 2 s ) z 2 ( y 1 + 2 s )( y 2 + 2 s ) − x 2 w 3 = x 2 z 1 − ( y 2 + 2 s ) z 2 2 s w 3 . (3.31) With tw o p ossibilities of the sign c hoice for w 3 in (3.29) we hav e that the in v erse image in the in tegral manifold { S = s, T = τ } ∩ O of a generic p oint ( x, ξ ) con ta ins 16 p oin ts. W e also need t he explicit form ula for w 3 in terms of x, ξ . F r o m (3.27) w e write y 1 + 2 s = 2 s 4[2 τ ξ − τ ( x 2 − τ ) + 4 s 2 χ 2 ] − ( p Φ 1 − p Φ 2 ) 2 + ( p Ψ 1 − p Ψ 2 ) 2 16 s 2 τ + ( p Ψ 1 − p Ψ 2 ) 2 , y 2 + 2 s = 2 s 4[2 τ ξ − τ ( x 2 − τ ) + 4 s 2 χ 2 ] − ( p Φ 1 + p Φ 2 ) 2 + ( p Ψ 1 + p Ψ 2 ) 2 16 s 2 τ + ( p Ψ 1 + p Ψ 2 ) 2 . (3.32) 8 Note that [16 s 2 τ + ( p Ψ 1 − p Ψ 2 ) 2 ][16 s 2 τ + ( p Ψ 1 + p Ψ 2 ) 2 ] = 64 s 2 ( τ ξ 2 + σ x 2 − τ σ ) . (3.33) Then D = p Φ 1 Φ 2 Ψ 1 Ψ 2 − P 2( τ ξ 2 + σ x 2 − τ σ ) , where P ( x, ξ ) = ξ 4 + 2 τ ξ 3 + 2[( τ − 2 s 2 − p 2 ) x 2 − τ (2 s 2 − p 2 ) − r 4 ] ξ 2 − − 8 s 2 [( τ − 2 χ 2 ) x 2 + τ χ 2 ] ξ − 4 s 2 ( x 2 − χ 2 )[2( τ − p 2 ) x 2 − ( τ 2 − r 4 )] . Let Q ( x, ξ ) = ( ξ + τ + 2 s 2 − p 2 ) 2 − 4 s 2 x 2 + r 4 − (2 s 2 − p 2 ) 2 , P 1 ( x, ξ ) = P ( x, ξ ) + 2 xQ ( x, ξ ) p τ ξ 2 + σ x 2 − τ σ , P 2 ( x, ξ ) = P ( x, ξ ) − 2 xQ ( x, ξ ) p τ ξ 2 + σ x 2 − τ σ . Direct calculation prov es the iden tit y P 1 P 2 ≡ Φ 1 Φ 2 Ψ 1 Ψ 2 . On the other hand, P 1 + P 2 ≡ 2 P . Therefore, D = − ( p P 1 − p P 2 ) 2 4( τ ξ 2 + σ x 2 − τ σ ) , whence w 3 = i p P 1 − p P 2 2 p 2 s ( τ ξ 2 + σ x 2 − τ σ ) . (3.34) Belo w w e use o ne more represen tatio n of w 1 , w 2 not dep ending on w 3 . The system (3.2) yields x 2 w 2 1 = − z 2 w 1 w 3 − ( y 2 + 2 s ) w 1 w 2 , x 1 w 2 2 = − z 1 w 2 w 3 − ( y 1 + 2 s ) w 1 w 2 . Substitute the expressions for w 1 w 3 , w 2 w 3 found from the in v erse prop ortion in (3.31) and eliminate w 1 w 2 b y (3.4) to obtain 2 s x 2 w 2 1 = − ( µ 1 − 2 sτ ) − 2 sx 2 , 2 s x 1 w 2 2 = − ( µ 2 − 2 sτ ) − 2 sx 2 . Then from (3.25) w e find w 1 = i 4 s √ x 2 ( p Θ 1 + p Θ 2 ) , w 2 = i 4 s √ x 1 ( p Θ 1 − p Θ 2 ) , (3.35) where Θ 1 ( x, ξ ) = ( ξ − 2 sx ) 2 − 4 s 2 χ 2 , Θ 2 ( x, ξ ) = ( ξ + 2 sx ) 2 − 4 s 2 χ 2 . (3.36) Note that the pro ducts Θ 1 Θ 2 and Ψ 1 Ψ 2 coincide. The c hoice of the signs in (3.35) is deter- mined b y the condition ( √ x 2 w 1 )( √ x 1 w 2 ) = x ξ / 2 s following from (3.8). The signs of the complex conjugate v alues √ x 1 , √ x 2 m ust b e chosen in suc h a w a y that the express ions (3.35), (3.35) satisfy one of the equations (3.2), (3.3) (then the other t w o hold automatically). Th us, all phase v aria bles are algebraically expressed in terms of tw o auxiliary v ariables x, ξ ; the domain of the latter dep ends on the constan ts of the first in tegrals. 9 4 Singularit ies of th e indu ced sympl ectic s tructure The Hamilto nia n structure of the system (1.1) is provided b y the Pois son brac kets on R 9 ( ω , α , β ). In notation (2.4) these brac k ets are [2] { M j , M k } = ε j k l M l , { M j , α k } = ε j k l α l , { M j , β k } = ε j k l β l , { α j , α k } = { α j , β k } = { β j , β k } = 0 . (4.1) Being restricted to P 6 they correspo nd to Kirillo v’s symplectic form λ ∈ Λ 2 ( P 6 ). Recall the follo wing we ll-kno wn facts. Let N ⊂ P 6 b e a submanifold defined in P 6 b y t w o indep enden t equations f 1 = 0 , f 2 = 0 (4.2) and let X 1 , X 2 b e the Hamiltonian v ector fields with the Hamilton functions f 1 , f 2 . Then the span of X 1 , X 2 at eac h p oin t ζ ∈ N is sk ew orthogo nal to the tangent space T ζ N . Therefore t he restriction of λ to N is non-degenerate at the p o in t ζ if and only if { f 1 , f 2 } ( ζ ) = λ ( X 1 , X 2 )( ζ ) 6 = 0 . (4.3) Let us calculate the bra c ket { R 1 , R 2 } of the functions (1.13). As R 1 is pure imag inary denote R = 1 i { R 1 , R 2 } . (4.4) The ch ange of v a riables (1.7) is linear with constan t co efficien ts, so the rules ( 4.1) are easily transformed for new co ordinates. Omitting tec hnical details w e presen t R in t he fo rm R = F 1 w 3 1 w 3 2 w 2 3 [ w 3 ( w 3 F 2 + F 3 ) 2 − F 4 ] , (4.5) where F 1 = z 1 z 2 w 3 + x 2 z 1 w 1 + x 1 z 2 w 2 , F 2 = w 1 w 2 w 3 + z 2 w 1 + z 1 w 2 , F 3 = 2 w 2 1 w 2 2 + x 2 w 2 1 + x 1 w 2 2 , F 4 = 4 w 2 1 w 2 2 [( w 2 1 w 2 2 + x 2 w 2 1 + x 1 w 2 2 ) w 3 + x 2 z 1 w 1 + x 1 z 2 w 2 ] . (4.6) The v ariables y 1 , y 2 ha v e b een eliminated as the solution of the system (1.12) y 1 = − w 1 w 3 ( x 2 w 1 + z 2 w 3 ) + F 1 w 1 w 2 w 3 , y 2 = − w 2 w 3 ( x 1 w 2 + z 1 w 3 ) + F 1 w 1 w 2 w 3 . (4.7) Fixing the v alues h, k , g , s, τ of the functions (1.3), (2.11), (2.12) we ha v e w 3 F 2 + F 3 = 2 w 1 w 2 h − 2( x 1 x 2 + z 1 z 2 − τ ) , F 4 = 4 w 2 1 w 2 2 w 3 ( τ − k ) . (4.8) Then from (2 .14), (3 .3), (3.4), (3.8), (3.9) w e obtain w 3 F 2 + F 3 = ξ s ( p 2 − τ 2 s − s ) , F 4 = − σ ξ 2 w 3 4 s 4 , F 1 = ξ w 3 , w 3 1 w 3 2 = ξ 3 8 s 3 . (4.9) Finally , the expression (4.5) ta k es the f o rm R = 8 s  s 4 − ( p 2 − τ ) s 2 + p 4 − r 4 4  . ( 4 .10) 10 Ob viously , R is a first integral of the induced system, and the equation s 4 − ( p 2 − τ ) s 2 + p 4 − r 4 4 = 0 (4.11 ) defines the set of the in tegral constants s, τ suc h t ha t on the corresp onding inte gral ma nif o lds the 2-form induced on O b y t he symplectic structure λ degenerates. As the functions (2.11), (2.12) ha v e an algebraic structure, the set (4.11) has co dimension 1 in O . Th us, λ | O is almost ev erywhere non-degenerate. T o lo cate the set (4.11) on the surface (2.1 4) in R 3 ( h, k , g ) notice that, in virtue of (2.14), ∂ ∂ s ( h, k , g ) × ∂ ∂ τ ( h, k , g ) =  s 4 − ( p 2 − τ ) s 2 + p 4 − r 4 4  ( τ − p 2 2 s 4 , 1 2 s 3 , 1 s 4 ) . Therefore, the equation (4.11) defines cuspidal edges of the surface (2.14). 5 Bifurcation diagram and the existenc e of motion s In tro duce the in tegral map J of t he dynamical system o n O J ( ζ ) = ( S ( ζ ) , T( ζ )) ∈ R 2 , ζ ∈ O . (5.1) The bifurcation diagram Σ( J ) of this map is, by definition, the set of pairs ( s, τ ) ov er whic h J is not lo cally trivial. In our case all in tegral manifolds (3.1) are compact. Therefore, Σ( J ) is the set of critical v alues of J . Define the admis s ible r e gion a s the image of the map (5.1), i.e., the set o f v alues ( s, τ ) suc h that the in tegral manifold (3.1) is not empt y . Obv iously , ∂ J ( O ) ⊂ Σ( J ). Due to the ab o v e results the existence o f a t r a j ectory o n O with give n s, τ is equiv alen t to the existence of a p oin t on the ( x, ξ )-plane f o r whic h the v alues (3.26)–(3.28), (3.35), (3.35) satisfy (3.7). It easily follows f rom (3.25), (3.28), (3.8) that the conditions (3.7), in turn, a re equiv alen t to the system o f inequalities Φ 1 ( x, ξ ) > 0 , Φ 2 ( x, ξ ) 6 0 , (5.2 ) Ψ 1 ( x, ξ ) > 0 , Ψ 2 ( x, ξ ) 6 0 (5.3) considered in the quadran t x > 0, s ξ > 0. Theorem 3. The bi f ur c ation di agr am Σ( J ) c onsists of the fol lowing subsets of the ( s, τ ) -plan e : 1 ◦ ) τ = ( a + b ) 2 , s ∈ [ − a, 0) ∪ [ b, + ∞ ); 2 ◦ ) τ = ( a − b ) 2 , s ∈ [ − a, − b ] ∪ (0 , + ∞ ); 3 ◦ ) s = − a, τ > ( a − b ) 2 ; 4 ◦ ) s = − b, τ > ( a − b ) 2 ; 5 ◦ ) s = b, τ 6 ( a + b ) 2 ; 6 ◦ ) s = a, τ 6 ( a + b ) 2 ; 7 ◦ ) τ = 0 , s ∈ (0 , + ∞ ); 8 ◦ ) τ = ( √ a 2 − s 2 + √ b 2 − s 2 ) 2 , s ∈ [ − b, 0) ; 9 ◦ ) τ = ( √ a 2 − s 2 − √ b 2 − s 2 ) 2 , s ∈ (0 , b ]; 10 ◦ ) τ = − ( √ s 2 − a 2 − √ s 2 − b 2 ) 2 , s ∈ [ a, + ∞ ) . 11 Theorem 4. The so l utions of the system (1.1) under the c onditions (1.1 2) exist iff the c onstants of the first i n te gr als (2.11) , (2.12) satisfy one of the fol low i n g 1 ◦ ) − a 6 s 6 − b, τ > ( a − b ) 2 ; 2 ◦ ) − b 6 s < 0 , τ > ( p a 2 − s 2 + p b 2 − s 2 ) 2 ; 3 ◦ ) 0 < s 6 b, τ 6 ( p a 2 − s 2 − p b 2 − s 2 ) 2 ; 4 ◦ ) b 6 s 6 a, τ 6 ( a + b ) 2 ; 5 ◦ ) s > a, − ( p s 2 − b 2 − p s 2 − a 2 ) 2 6 τ 6 ( a + b ) 2 . The complete pro of of these statemen ts is purely techn ical (see [1 0 ]) and con tains the scrupu- lous a nalysis o f the regions on the ( x, ξ )-plane defined by (5 .2), (5.3) and the cases of their structural transformatio ns. Anot her a pproac h is suggested in [1 1] for t he pair ( S, H ). It is based on the classification of the tra jectories in O satisfying the condition rank ( H × K × G ) < 2. Figure 1: The admissible regio n in the ( s, τ )- plane. The admissible region is shaded in Fig.1. The dense lines and curv es represen t t he equations of the bifurcat io n diagram, the dashed curv e illustrates the equation (4.11), i.e., the first in tegrals constan ts suc h that the symplectic structure is degenerate on the corresp onding in tegral manifolds. 6 Separation of v ariables In the sequel w e supp ose that τ σ 6 = 0. In fact, if τ = 0, then from (2.14) w e obtain the relation (2 g − p 2 h ) 2 = r 4 k c har acteristic for the critical manifold N [9]. The equations of motion on N w ere explicitly inte grated in [8]. If σ = 0, then the equations (2.14 ) yield one of the relations (1.11). This case corresp onds to t he set of p oints (1.9). At these p oin ts the manifold O fails to b e smo oth. The corresp onding tra jectories are the p endulum motions (1.10). Considering the second equation (3.21) denote µ = | r 2 x 1 − τ y 1 | = | r 2 x 2 − τ y 2 | . Then µ 2 = τ ξ 2 + σ x 2 − τ σ . (6 .1) This equation defines t he second-order surface M in three-dimensional space R 3 ( x, ξ , µ ). Each tra jectory in O is in a natural wa y represen ted b y a curve on M . 12 Theorem 5. Supp osing τ σ 6 = 0 , i n tr o duc e the variables U = τ ξ + xµ √ σ ( τ − x 2 ) , V = τ ξ − xµ √ σ ( τ − x 2 ) . (6.2) Then the e quations of motion on O sep ar ate dU p Q ( U ) − dV p Q ( V ) = 0 , U dU p Q ( U ) − V dV p Q ( V ) = dt p 2 sτ σ . (6.3) Her e Q ( w ) = ( w 2 − 1)( σ w 2 − 4 s 2 χ 2 )[( √ σ w + τ ) 2 − r 4 ] . (6.4) Pr o of. Consider the lo cal co ordinates u, v on the surface M : ξ = √ σ uv + 1 u + v , x = √ τ u − v u + v , µ = √ τ σ uv − 1 u + v . (6.5) In a ddition to (3 .1 9) note the following tw o iden tities for the constants in tro duced ab ov e, σ + 2 τ ( p 2 ± r 2 ) = ( τ ± r 2 ) 2 . (6.6) The p olynomials (3.15), (3.24), (3.36 ) b ecome Φ 1 = κ ϕ 1 ( u ) ϕ 1 ( v ) , Φ 2 = κ ϕ 2 ( u ) ϕ 2 ( v ) , Ψ 1 = κ ψ 1 ( u ) ψ 2 ( v ) , Ψ 2 = κ ψ 2 ( u ) ψ 1 ( v ) , Θ 1 = κ θ 2 ( u ) θ 1 ( v ) , Θ 2 = κ θ 1 ( u ) θ 2 ( v ) , where κ = 1 / ( u + v ) 2 and ϕ 1 ( w ) = √ σ (1 + w 2 ) + 2( τ + r 2 ) w , ϕ 2 ( w ) = √ σ (1 + w 2 ) + 2( τ − r 2 ) w , ψ 1 ( w ) = 2 s [( χ + √ τ ) w 2 − ( χ − √ τ )] , ψ 2 ( w ) = 2 s [( χ − √ τ ) w 2 − ( χ + √ τ )] , θ 1 ( w ) = √ σ (1 − w 2 ) + 4 s √ τ w , θ 2 ( w ) = √ σ (1 − w 2 ) − 4 s √ τ w . Then from (3 .26)–(3.28) w e find the expressions fo r the configuration v a riables x 1 = 2 sτ r 2 " p ϕ 1 ( u ) ϕ 2 ( v ) + p ϕ 2 ( u ) ϕ 1 ( v ) p θ 1 ( u ) θ 1 ( v ) − p θ 2 ( u ) θ 2 ( v ) # 2 , x 2 = 2 sτ r 2 " p ϕ 1 ( u ) ϕ 2 ( v ) − p ϕ 2 ( u ) ϕ 1 ( v ) p θ 1 ( u ) θ 1 ( v ) + p θ 2 ( u ) θ 2 ( v ) # 2 , (6.7) y 1 = 2 s h p ϕ 1 ( u ) ϕ 2 ( v ) + p ϕ 2 ( u ) ϕ 1 ( v ) i 2 − 4 σ ( u v − 1) 2 h p θ 1 ( u ) θ 1 ( v ) − p θ 2 ( u ) θ 2 ( v ) i 2 , y 2 = 2 s h p ϕ 1 ( u ) ϕ 2 ( v ) − p ϕ 2 ( u ) ϕ 1 ( v ) i 2 − 4 σ ( uv − 1) 2 h p θ 1 ( u ) θ 1 ( v ) + p θ 2 ( u ) θ 2 ( v ) i 2 , (6.8) z 1 = 1 2 r ( u + v ) h p ϕ 1 ( u ) ϕ 1 ( v ) + p ϕ 2 ( u ) ϕ 2 ( v ) i , z 2 = 1 2 r ( u + v ) h p ϕ 1 ( u ) ϕ 1 ( v ) − p ϕ 2 ( u ) ϕ 2 ( v ) i . (6.9) 13 Hence, in particular, √ x 1 = p 2 s τ r p ϕ 1 ( u ) ϕ 2 ( v ) + p ϕ 2 ( u ) ϕ 1 ( v ) p θ 1 ( u ) θ 1 ( v ) − p θ 2 ( u ) θ 2 ( v ) , √ x 2 = p 2 s τ r p ϕ 1 ( u ) ϕ 2 ( v ) − p ϕ 2 ( u ) ϕ 1 ( v ) p θ 1 ( u ) θ 1 ( v ) + p θ 2 ( u ) θ 2 ( v ) . (6.10) Here the arbitrary c hoice o f sign is provided by the algebraic v alue √ 2 sτ . F ro m (3.35) w e hav e √ x 2 w 1 = i 4 s p θ 2 ( u ) θ 1 ( v ) + p θ 1 ( u ) θ 2 ( v ) u + v , √ x 1 w 2 = i 4 s p θ 2 ( u ) θ 1 ( v ) − p θ 1 ( u ) θ 2 ( v ) u + v . (6.11) Substitute (6.10) in to (6.11) t o obtain the expressions f o r the v ariables defining the equatorial comp onen ts of the angular v elo cit y w 1 = ir 4 s √ 2 sτ h p θ 2 ( u ) θ 1 ( v ) + p θ 1 ( u ) θ 2 ( v ) i h p θ 1 ( u ) θ 1 ( v ) + p θ 2 ( u ) θ 2 ( v ) i ( u + v ) h p ϕ 1 ( u ) ϕ 2 ( v ) − p ϕ 2 ( u ) ϕ 1 ( v ) i , w 2 = ir 4 s √ 2 sτ h p θ 2 ( u ) θ 1 ( v ) − p θ 1 ( u ) θ 2 ( v ) i h p θ 1 ( u ) θ 1 ( v ) − p θ 2 ( u ) θ 2 ( v ) i ( u + v ) h p ϕ 1 ( u ) ϕ 2 ( v ) + p ϕ 2 ( u ) ϕ 1 ( v ) i . (6.12) The axial comp onen t is f o und fr o m (3.35), w 3 = i 2 √ 2 sτ σ p θ 1 ( u ) θ 2 ( u ) ϕ 1 ( v ) ϕ 2 ( v ) − p ϕ 1 ( u ) ϕ 2 ( u ) θ 1 ( v ) θ 2 ( v ) ( u + v )( u v − 1) . ( 6 .13) Th us w e hav e express ed a ll phase v aria bles in terms of u, v . T o o bta in t he differential equations for u, v consider the follow ing v ariables s 1 = x 2 + z 2 + r 2 2 x , s 2 = x 2 + z 2 − r 2 2 x . (6.14) The deriv ativ es, in virtue of t he system (1.8), a re s ′ 1 = r 2 4 x 3 ( z 1 + z 2 )( x 1 w 2 − x 2 w 1 ) , s ′ 2 = r 2 4 x 3 ( z 1 − z 2 )( x 1 w 2 + x 2 w 1 ) . (6.15) On the other hand, from (6.14), (6.5) w e ha v e s 1 = √ σ ( uv + 1) + ( τ + r 2 )( u + v ) 2 √ τ ( u − v ) , s 2 = √ σ ( uv + 1) + ( τ − r 2 )( u + v ) 2 √ τ ( u − v ) , whence ∂ s 1 ∂ u = − ϕ 1 ( v ) 2 √ τ ( u − v ) 2 , ∂ s 1 ∂ v = ϕ 1 ( u ) 2 √ τ ( u − v ) 2 , ∂ s 2 ∂ u = − ϕ 2 ( v ) 2 √ τ ( u − v ) 2 , ∂ s 2 ∂ v = ϕ 2 ( u ) 2 √ τ ( u − v ) 2 . 14 Therefore, du dt = 2 √ τ ( u − v ) 2 ϕ 1 ( u ) ϕ 2 ( v ) − ϕ 2 ( u ) ϕ 1 ( v ) [ ϕ 2 ( u ) ds 1 dt − ϕ 1 ( u ) ds 2 dt ] , dv dt = 2 √ τ ( u − v ) 2 ϕ 1 ( u ) ϕ 2 ( v ) − ϕ 2 ( u ) ϕ 1 ( v ) [ ϕ 2 ( v ) ds 1 dt − ϕ 1 ( v ) ds 2 dt ] . (6.16) Substitute the v alues (6.9) –(6.11) into (6.15) and the resulting expressions in to (6 .16). W e obtain f ( u, v ) du dt = p ϕ 1 ( u ) ϕ 2 ( u ) θ 1 ( u ) θ 2 ( u ) 2 u p 2 s τ σ , f ( u, v ) dv dt = p ϕ 1 ( v ) ϕ 2 ( v ) θ 1 ( v ) θ 2 ( v ) 2 v p 2 s τ σ , (6.17) where f ( u, v ) = ( u − v )(1 − uv ) uv =  v + 1 v  −  u + 1 u  . The v ariables (6.2) in terms of u, v hav e the fo rm U = 1 2  u + 1 u  , V = 1 2  v + 1 v  . (6.18) T o b e definite choose u = U − p U 2 − 1, v = V − p V 2 − 1. Then the equations (6.17) yield ( U − V ) dU dt = 1 √ 2 s τ σ p Q ( U ) , ( U − V ) dV dt = 1 √ 2 s τ σ p Q ( V ) (6.19) with the p olynomial (6 .4) of degree 6. This system is obvious ly equiv alent to the system (6.3); the la tter hav e the standard form for hy p erelliptic quadratur es. T o rev eal the connection of Theorem 5 with the bifurcation diagram supp ose that the p oly- nomial (6.4) has a multiple ro ot. The r esultant of Q ( w ) and Q ′ ( w ) is 2 32 a 4 b 4 ( a 2 − b 2 ) 2 s 10 τ 12 σ 8 χ 2 ( a 2 − s 2 ) 2 ( b 2 − s 2 ) 2 . (6.20) According to (2.14), s 6 = 0 on O . The equation σ = 0 g iv es τ = ( a ± b ) 2 . The case χ = 0 provide s the v alues τ = ( p a 2 − s 2 ± p b 2 − s 2 ) 2 . Thus , the bifurcatio n diagram o f J = S × T b elongs to the discriminan t set o f the p olynomial (6.4). This fact is t ypical for systems with separating v ariables. Note that all ro ots o f the p olynomial (6.4) are explicitly expresse d in terms of the in tegral constan ts a long with the ro ots of the p o lynomial ϕ 1 ( w ) ϕ 2 ( w ) θ 1 ( w ) θ 2 ( w ) (6.21) defining the solutions of the system (6 .17). The v ar ia bles in tro duced in this section a re r eal only in the case τ > 0 and σ > 0. F or ot her combinations of signs the definition (6.5) of the lo cal co ordinates u , v needs o b vious mo difications. Then the interv als of o scillations for the v ariables U, V and u , v are easily determined for any giv en v a lues of s, τ or, more exactly , for each connected comp o nen t o f R 2 \ Σ( J ). The obtained differen tial equations and the explicit formulae for the phase v ariables in terms of u, v provide the p o ssibilit y to in v estigate the phase top ology of the case considered and to calculate analytically the n umerical in v aria n ts of the corresp onding Liouville foliation. 15 References [1] M. P . Kharlamo v. Bifurcation diagra ms of the Kow alevski top in t w o constan t fields. R e gul. Chaotic D yn. , 10 (200 5), 381–39 8 . [2] O. I. Bogoy avle nsky . Euler equations on finite-dimension Lie algebras arising in phys ical prob- lems. Commun. Math. Phys. , 95 (198 4), 307–31 5 . [3] A. G. Reyman, M. A. Semeno v-Tian-Shansky . Lax represen tation with a sp ectral parameter for the Kow alewski top and its generalizations. L ett. Math. Ph ys. , 14 (1987), 55 –61. [4] M. P . Kharlamo v. Critical set and bifurcation diagram of the pro blem of motion of the Ko w a levski top in do uble field. Me ch. Tver d. T el a , N 34, 2004, 47 –58. (In R ussian) [5] S. Kow alevski. Sur le probl` eme de la rotation d’un cor ps solide autour d’un p oint fixe. A cta Math. , 12 (188 9), 177 –232. [6] G. G. App elro t . Non-completely symmetric hea vy gyroscop es. In: Motion of a rigid b o dy ab out a fixe d p oint . Collection of pap ers in memory of S.V.Kov alevsk ay . Acad. Sci. USSR, Mosco w- Leningrad, 1940, 61–15 6 . (In Russian) [7] D. B. Z o tev. F o menk o-Z iesc hang inv ariant in the Bogoy av lenskyi case. R e gul. Chaotic Dyn. , 5 (2000), 4 3 7–458. [8] M. P . Kharlamo v, A. Y. Sa vush kin. Separat io n of v ariables and integral ma nif o lds in one par - tial problem of motion o f the generalized Ko w alevski top. Ukr. Math. Bul l. , 1 (2004), 548–565 . [9] M. P . Kharlamo v. One class of solutions with tw o in v ar ian t relat io ns in the problem of motion of the Kow alevsky top in double constan t field. Mekh. tver d. tela , N 32 , 2002 , 3 2–38. (In Russian) [10] M. P . Kharlamov . Bifurcation diagram of the generalized 4th App elrot class. Mekh. tver d. tela , N 35, 2005 , 3 8–48. (In Russian) [11] M. P . Kharlamov , E. G. Sh v edo v. On the existence of motions in the generalized 4th App elrot class. R e gul. Chao tic Dyn. , 11 (2 0 06), 33 7–342. 16