on one integrable system with a cubic first integral

DOI 10.1007/s11005-012-0562-9Lett Math Phys (2012) 101:143–156

On One Integrable System With a Cubic FirstIntegral

ALEXANDER VLADIMIROVICH VERSHILOV andANDREY VLADIMIROVICH TSIGANOVSt. Petersburg State University, St. Petersburg, Russia.e-mail: [email protected]; [email protected]

Received: 7 March 2011 / Revised: 20 March 2012 / Accepted: 1 April 2012Published online: 21 April 2012 – © Springer 2012

Abstract. Recently, Valent studied one integrable model with a cubic first integral of motionusing a special coordinate system. We describe the bi-Hamiltonian structures and variablesof separation for this system.

Mathematics Subject Classification. 70H20, 70H06, 37K10.

Keywords. integrable systems, bi-Hamiltonian geometry, trigonal curve.

1. Introduction

A few years ago, Selivanova [10] and Kiyohara [9] proved some existence theo-rems for the Riemannian metrics on two-dimensional sphere having the propertyof the geodesic flows admiting first integrals of motion, which are homogeneousthird and higher order polynomials. Valent [15] gave the explicit form of the cubicintegrals associated with the solution of a nonlinear third-order ordinary differen-tial equation proposed by Selivanova. The main point which stems from the Valentwork is that the local coordinates choice is of the utmost delicacy, since it deter-mines the structure of the differential equations to be solved eventually.

The aim of this note is to consider the Valent models defined by a naturalHamilton function

H = T + V =2∑

i, j=1

gi j (q2) pi p j + V (q1,q2) (1.1)

with metric g depending on one variable, and cubic additional integral of motionwith the leading terms

H2 = p p31 +2q T p1 +· · · , p ∈R, q ≥0. (1.2)

According to [15], relevant metrics are described by a finite number of parametersand lead to a large class of models mainly on the manifolds S

2 and H2. By suit-

able choices of the parameters entering the construction these systems are globally

144 ALEXANDER VLADIMIROVICH VERSHILOV AND ANDREY VLADIMIROVICH TSIGANOV

defined and contain special cases as the known systems of Goryachev–Chaplygin,Goryachev, and Dullin and Matveev.

In [14,17], we introduce a natural Poisson bivector depending on two arbitraryfunctions, which allows us to describe similar family of integrable system withcubic additional integral of motion in spherical coordinates. However, because achoice of appropriate coordinates plays a crucial role in the transformation ofthe Selivanova existence theorem to the Valent constructive approach, we want toobtain similar Poisson bivector in the Valent coordinate system. It is the maindifference with [14,17]. Then, we calculate the corresponding variables of separa-tion and prove that equations of motion are linearized on strata of a genus threenon-hyperelliptic curve. It may allow us to get non-meromorphic solutions of theValent equations of motion on the manifolds S

2 and H2 in term of the Weierstrass

σ -functions [2].

2. Settings

In this section, we recall some necessary facts about natural bi-integrable sys-tems on Riemannian manifolds admitting separation of variables in the Hamilton–Jacobi equation [8,13,14]. All constructions are local and depend on a suitablechoice of the local coordinates.

Let Q be a n-dimensional Riemannian manifold. Its cotangent bundle T ∗Q isnaturally endowed with canonical invertible Poisson bivector P , which has a stan-dard form in fibered coordinates z = (q1, . . . ,qn, p1, . . . , pn) on T ∗Q

P =(

0 I−I 0

), { f, g}=〈P d f,dg〉=

2n∑

i=1

Pi j∂ f

∂zi

∂g

∂z j. (2.1)

To calculate the variables of separation for the given integrable system with inte-grals of motion H1, . . . , Hn in involution

{Hi , Hj }=0, i, j =1, . . . ,n,

in the bi-Hamiltonian set-up, we have to solve equations

[P, P ′]= [P ′, P ′]=0, (2.2)

where [., .] means a Schouten bracket, and

{Hi , Hj }′ =0, i, j =1, . . . ,n, { f, g}′ = 〈P ′ d f,dg〉, (2.3)

with respect to the Poisson bivector P ′. Then, we have to calculate the so-calledNijenhuis operator (or hereditary, or recursion operator)

N = P ′ P−1. (2.4)

ONE INTEGRABLE SYSTEM WITH A CUBIC FIRST INTEGRAL 145

If N has, at every point, the maximal number of different functionally independenteigenvalues u1, . . . ,un , then they may be treated either as action variables (integralsof motion) or as variables of separation for this dynamical system [5,8,13].

Separation of variables for natural integrable systems with higher order inte-grals of motion always involves generic canonical transformation of the wholephase space. The definition of the natural Hamiltonians (1.1) is non-invariant withrespect to such transformations of coordinates on the whole phase space.

In case when habitual objects (geodesics, metrics and potentials) lose their geo-metric sense, and the remaining invariant equations (2.2, 2.3) have infinite numberof solutions, the notion of natural Poisson bivectors has become de-facto a veryuseful practical tool for the calculation of variables of separation [13,14].

Similar to the natural Hamilton function on T ∗Q, the natural Poisson bivectorP ′ is a sum of the geodesic Poisson bivector P ′

T and the potential Poisson bivec-tor defined by a torsionless (1,1) tensor field �(q1, . . . ,qn) on Q associated withpotential V [13,14]:

P ′ = P ′T +

⎛

⎜⎝0 �i j

−� j i

n∑

k=1

(∂�ki

∂q j− ∂�k j

∂qi

)pk

⎞

⎟⎠ . (2.5)

The geodesic Poisson bivector P ′T is defined by n ×n geodesic matrix � on T ∗Q:

P ′T =

⎛

⎜⎜⎜⎜⎝

n∑

k=1

x jk(q)∂� jk

∂pi− yik(q)

∂�ik

∂p j�i j

−� j i

n∑

k=1

(∂�ki

∂q j− ∂�k j

∂qi

)zk(p )

⎞

⎟⎟⎟⎟⎠. (2.6)

In fact, for the given matrix � functions x, y and z are completely determined bythe equations

[P, P ′T ]= [P ′

T , P ′T ]=0, (2.7)

whereas � is obtained as a solution of the Equation (2.2).In fact, herein, we suppose that the separation of the geodesic motion is a nec-

essary condition for the separation at V �=0 and, therefore, we use the terms geo-desic bivector defined by geodesic matrix � and the potential bivector defined bypotential matrix �. Recall that matrices � and � are not tensor fields with respectto generic canonical transformations of the whole phase space and we do not havean invariant geometric theory behind this construction. Discussion of this usefulanzats (2.5) may be found in [13,14].

3. Natural Poisson Bivectors for the Valent Systems

In [15], solving the equation

{H1, H2}=0


in the framework of the Laplace method, author prefers to use special canonicalcoordinates

q1 =φ, q2 = ζ, p1 = pφ, p2 = pζ , (3.8)

Here H1 is a natural Hamilton function (1.1) with diagonal metric g(ζ ) and H2 isfixed by (1.2).

According to [15], special case q = 0 in (1.2) is rather difficult to obtain as thelimit of the general case q �= 0. So, we first integrate the special case at q = 0 andonly then we consider the generic case. In contrast with [15], we restrict ourselvesonly by local analysis.

3.1. CASE q =0

Let us take integrals of motion from [15], Theorem 1 formulae (14–15):

H (0)

1 = 12

(F p2

ζ + G

4Fp2φ

)+λ

√F cosφ +μζ, (′= Dζ )

H (0)

2 = p3φ −2λ

(√F sin φpζ + (

√F)′ cosφpφ

)−2μpφ,

(3.9)

with

F =−2ρ0 +3c0ζ + ζ 3, G =9c20 +24ρ0ζ −18c0ζ

2 −3ζ 4. (3.10)

Here λ,μ,ρ0 and c0 are arbitrary parameters.

PROPOSITION 1. Integrals of motion H (0)

1,2 (3.9) are in bi-involution

{H (0)

1 , H (0)

2 }={H (0)

1 , H (0)

2 }′ =0 (3.11)

with respect to a pair of compatible Poisson brackets associated with the canonicalPoisson bivector P (2.1) and natural Poisson bivector P ′ (2.5) defined by

�=⎛

⎝ 0i2

(∂

∂ζ+ F ′

F

)F

0 F

⎞

⎠ , F=−(

2pζ − iF ′

Fpφ

)2

(3.12)

and

�=−4λe−iφ√

F

(1 00 1

), x22 =− F

F ′ , (3.13)

whereas other functions xi j , yi j and zk in (2.6) are equal to zero.

Below, in order to extend the known palette of natural Poisson bivectors listedin [14], we consider another solution of the Equation (2.2) depending on both vari-ables φ and ζ .


PROPOSITION 2. Integrals of motion H (0)

1,2 (3.9) are in the bi-involution (3.11) withrespect to canonical Poisson bracket and bracket {., }′ associated with the naturalPoisson bivector P ′ (2.5) defined by 2×2 geodesic matrix

�=⎛

⎝ F − iF′

20 0

⎞

⎠ , F=[

eiφ(

iF ′

2√

Fpφ +√

F pζ

)]2

, (3.14)

diagonal potential matrix

�=λeiφ√

F

(1 00 1

)(3.15)

and function

y11 =− i2.

Other functions yik , xik and zk in the definition (2.6) equal zero.

As for the Lagrange top, this geodesic matrix (3.14) is factorized �=e2iφ �(ζ ), seediscussion in [14].

We are now able to analyze the corresponding recursion operator N = P ′ P−1.For instance, we can define the variables of separation u1,2 as roots of the linearin momenta polynomial

B(η)= (η−u1)(η−u2)=η2 − i√

Fη+ �11

=η2 − ieiφ

(3i(c0 + ζ 2)

2√−2ρ0 +3c0ζ + ζ 3

pφ +√

−2ρ0 +3c0ζ + ζ 3 pζ

)η

+λ

√−2ρ0 +3c0ζ + ζ 3eiφ, (3.16)

so that the characteristic polynomial of N reads as det(N −ηI)= (η−u21)

2(η−u22)

2.The conjugated momenta are equal to

puk = i pφ

uk− iλζ

u2k

, k =1,2.

The inverse transformation looks like

ζ =− iu1u2(u1 pu1 −u2 pu2)

λ(u1 −u2), pφ = i(u2

1 pu1 −u22 pu2)

u1 −u2,

φ = i2

ln

(λ2 F

u21u2

2

), pζ =− iF ′

2Fpφ − iλ(u1 +u2)

u1u2.

(3.17)

In these variables of separation, our initial bivector P ′ (3.12, 3.13) looks like

P ′ =3

⎛

⎜⎜⎝

0 0 u21 p2

u10

0 0 0 u22 p2

u2−u21 p2

u10 0 0

0 −u22 p2

u20 0

⎞

⎟⎟⎠ .


Now, we can calculate the integrals of motion

H (0)

1 =− iμu1u2(u1 pu1 −u2 pu2)

λ(u1 −u2)+ (u2

2 +u1u2 +u21)ρ0λ

2

u21u2

2

−3ic0(u1 pu2 − pu1 u2)λ

2(u1 −u2)+ u1u2

2

− i(u21 p2

u1+u2 pu2 u1 pu1 +u2

2 p2u2

)(u1 pu1 −u2 pu2)u1u2

2(u1 −u2)λ,

H (0)

2 =−2iμ(pu1 u21 − pu2 u2

2)

u1 −u2+ 2(u1 +u2)ρ0λ

3

u21u2

2

+3ic0(pu1 − pu2)λ2

u1 −u2+ u1 +u2

λ− i(p3

u1u4

1 −u42 p3

u2)

u1 −u2,

and the control matrix

F =(

0u1u2

2λ−2λ u1 +u2

), defined by P ′d H (0)

i = P2∑

j=1

Fi j dH (0)j , i =1,2.

According to [5], the suitable normalized left eigenvectors of control matrix F formthe Stackel matrix S, so the notion of F allows us to compute the correspondingseparated relations

2∑

j=1

Si j H j +Ui =0, i =1,2.

Here functions Si j and Ui depend only on one pair (ui , pui ) of canonical variablesof separation [11].

In our case, the integrals of motion and the variables of separation are relatedvia the following separated relations

(u, z)= z3 + (3λ2c0 −2μu2)z +λu4 − H (0)

2 u3 +2λH (0)

1 u2 −2λ3ρ0 =0, (3.18)

at u =u1,2 and z = iu21,2 p1,2. Equation (u, z)=0 defines a genus three non-hyper-

elliptic curve C with the following base of the holomorphic differentials

�1 = du

2μu2 −3λ2c0 −3z2, �2 = udu

2μu2 −3λ2c0 −3z2,

�3 = zdu

2μu2 −3λ2c0 −3z2.

It is a non-hyperelliptic curve, because there are no birational transformations ofthe coordinates (z,u) which takes the curve to a hyperelliptic form. We can easyprove this fact using a modern computer algebra system. According to [3,4], it isthe so-called (3,4) trigonal curve.


Equations of motion in the variables of separation have the following form

u1

2μu21 −3λ2c0 −3z2

1

+ u2

2μu22 −3λ2c0 −3z2

2

= i2λ

, zk = iu2k pk,

u1u1

2μu21 −3λ2c0 −3z2

1

+ u2u2

2μu22 −3λ2c0 −3z2

2

=0.

Third differential enters into the following expression

z1u1

2μu21 −3λ2c0 −3z2

1

+ z2u2

2μu22 −3λ2c0 −3z2

2

= iu1u2(z1 − z2)

2λ(u1 −u2).

The two afore-mentioned quadratures in the integral form look like

P1∫

∞

(�1

�2

)+

P2∫

∞

(�1

�2

)=

(w1

w2

), Pi = (ui ,μi )∈C, (3.19)

here w1 = i2λ

t +β1 and w2 =β2. The full Abel map

P1∫

∞�+

P2∫

∞�+

P3∫

∞�=w =

⎛

⎝w1

w2

w3

⎞

⎠ , �=⎛

⎝�1

�2

�3

⎞

⎠ ,

can be inverted by means of the theta or sigma-function of the trigonal curveC . According to [2] here w3 is a transcendental function of w1 and w2, whereasquadratures (3.19) defines a map of the symmetric product C × C to a codimen-sion one subvariety of the Jacobian variety of C given analytically by the condi-tion σ(w)=0. This subvariety is so-called Wirtinger stratum discussed in [2,3] andσ is a sigma-function of the trigonal curve C .

3.2. CASE q �=0

Let us take integrals of motion from [15], Theorem 5 formulae (39-40):

H1 = 12ζ

(Fp2

ζ + G

4Fp2φ

)+

√F

2qζcosφ + β0

2qζ,

H2 = p p3φ +2q H1 pφ −√

F sin φ pζ − (√

F)′ cosφ pφ,

(3.20)

with

F = c0 + c1ζ + c2ζ2 + c3ζ

3, G = F ′2 −2F F ′′, c3 = p

q, (3.21)

where β0, c0, c1, c2 and p,q are arbitrary parameters.

PROPOSITION 3. After substitution of the new function F (3.21) into the previousdefinitions (3.12, 3.13) and (3.14, 3.15) one gets two compatible Poisson bivectors P ′and P ′, so that the integrals of motion H1,2 (3.20) are in the involution with respectto the corresponding Poisson brackets.


Other calculations are standard. Namely, the variables of separation are given by

B = (η−u1)(η−u2)=η2 − eiφ(

iF ′

2√

Fpφ +√

F pζ

)η−

√Feiφ

2q,

puk =− ipφ

uk− ζ

2qu2k

,

(3.22)

whereas the inverse transformation reads as

ζ = 2qu1u2(u1 pu1 −u2 pu2)

u1 −u2, pφ = i(u2

1 pu1 −u22 pu2)

u1 −u2,

φ = i2

ln

(F

4q2u21u2

2

)−π, pζ =− iF ′

2Fpφ − u1 +u2

2qu1u2.

(3.23)

In variables of separation integrals of motion (3.20) read as

H1 = 12

c3(u21 p2

u1+u1u2 pu1 pu2 +u2

2 p2u2

)

− c3

4(u1 pu1 −u2 pu2)

((u1 −u2)

p+ (u1 p2

u1−u2 p2

u2)c2

p

+ (u1 pu2 −u2 pu1)c1c3

2p2u1u2− (u1 −u2)β0c3

p2u1u2− (u1 −u2)(u2

1 +u1u2 +u22)c0c2

3

4p3u31u3

2

),

H2 = ip(u1 pu1 +u2 pu2)u1u2 pu1 pu2

− i2(u1 pu1 −u2 pu2)

(pu1 pu2(u

21 −u2

2)c1c3

2pu1u2+ (c2 pu1 pu2 −1)(pu1 − pu2)u1u2

− (u21 pu1 −u2

2 pu2)β0c3

pu1u2− (u4

1 pu1 −u42 pu2)c0c2

3

4p2u31u3

2

). (3.24)

Then we have to calculate the control matrix F and the corresponding Stackelmatrix S to get the desired separated relations, which are obtained by substitutingu = i u1,2 and z = p u2

1,2 pu1,2 in the following equation

=2z3 − c2 z2 − (8q2 H1u2 − c1)p

2qz − p2u4

−2ip2 H2u3 − β0 p2u2

q− p2c0

4q2=0. (3.25)

This equation (u, z) = 0 defines the genus three (3,4) algebraic curve with theholomorphic differentials

�1 = du

−2c3∂/∂z, �2 = udu

−2c3∂/∂z, �3 = zdu

−2c3∂/∂z. (3.26)


As above, these are easy calculations on any modern computer algebra systems.The desired quadratures read as

uu1

8p2 H1u21 −12c3z2

1 +4c2c3z1 − c1c23

+ uu2

8p2 H1u22 −12c3z2

2 +4c2c3z2 − c1c23

=0,

zk = p u2k pk,

andz1u1

8p2 H1u21 −12c3z2

1+4c2c3z1 − c1c23

+ z2u2

8p2 H1u22 −12c3z2

2 +4c2c3z2 − c1c23

=− p

8.

The afore-mentioned quadratures in the integral formu1∫

u0

�2 +u2∫

u0

�2 =β1,

u1∫

u0

�3 +u2∫

u0

�3 =− p

8t +β2, (3.27)

represent a part of the Abel-Jacobi map associated to the Wirtinger stratum of thegenus three non-hyperelliptic curve defined by (3.25) [2]. If we change u → iu thatcorresponds to transformation P ′ →−P ′, we gets the equation (u, z) (3.25) withreal coefficients, but the coefficient before time variable in (3.19) becomes an imag-inary number.

An explicit solution of such Jacobi inversion problems on strata of the trigonalJacobians in terms of the Weierstrass σ -functions is discussed in [2]. The resultsreferring to Weierstrass theory for the general trigonal curve of genus three maybe found in [3,4]. On the other hand, for a non-hyperelliptic Riemann surface ofgenus three the moduli space of rank two bundles compactifies into a singular pro-jective variety, which is closely related to the Kummer variety and which is a quar-tic hypersurface of P

7. We can try to apply the generic theory of vector bundles onalgebraic varieties to the corresponding integrable systems [3,16].

4. Possible Generalizations

Let Q be a 2-dimensional Riemannian manifold, and its cotangent bundle T ∗Q isnaturally endowed with four canonical coordinates q1,q2, p1 and p2.

According to [14] geodesic matrix � (3.12) belongs to a whole family of matri-ces � labelled by two arbitrary functions

�=⎛

⎝ 0 − i2

(∂

∂q2+2h(q2)

g(q2)

)F

0 F

⎞

⎠ , F=(

g(q2)p2 − ih(q2)p1

)2, i=√−1.

(4.28)

In this case,

�=α exp(

iq1 −∫

h(q2)

g(q2)dq2

)(1 00 1

), α ∈C. (4.29)


Here g(q2) and h(q2) are arbitrary functions and

x22 =− g(q2)

2h(q2).

Other functions yik , xik and zk in the definition (2.6) equal zero.A characteristic polynomial of the corresponding recursion operator N = P ′ P−1

reads as

B(η)=det(N −ηI)=(η2 − (F+2�11) η+�2

11

)2.

Here F (4.28) is a complete square, it allows us to introduce the coordinates v1,2

as roots of the linear in momenta p1,2 polynomial

B(λ)= (λ−v1)(λ−v2)=λ2 − i√

Fλ+�11. (4.30)

In this coordinates, B= (η+v21)2(η+v2

2)2. We prefer to use the linear in momentapolynomial B(λ) instead of B(η) because in this case it is easy to find a solutionof the equations

{B(λ), A(μ)}= λ

μ−λ

(B(λ)

λ− B(μ)

μ

), {A(λ), A(μ)}=0, (4.31)

with respect to other linear in momenta polynomial

A(λ)=∫

idq2

g(q2)− ip1

λ.

Equations (4.31) entail that canonically conjugated momenta are equal to

pv1,2 = A(λ=v1,2), (4.32)

up to canonical transformations pvk → pvk + φk(vk). In bi-Hamiltonian geometry,the eigenvalues of the recursion operator and the corresponding momenta are con-sidered either as integrals of motion or as variables of separation [5].

In the book [7], when Jacobi invented elliptic coordinates and successfully appliedthem to solve several important mechanical problems, such as the problem of geo-desic motion on an ellipsoid, and the problem of planar motion in a force field oftwo attracting centers, he wrote “The main difficulty in integrating a given differ-ential equation lies in introducing convenient variables, which there is no rule forfinding. Therefore, we must travel the reverse path and after finding some notablesubstitution, look for problems to which it can be successfully applied.”

Namely, if we have canonical variables of separation v1, v2, pv1 and pv2 we canalways substitute them into any possible separated relations

i (vi , pvi , H1, H2)=0 i =1,2 such that det[

∂i

∂Hj

]�=0.

Solving the resulting equations with respect to H1 and H2, we obtain function-ally independent integrals of motion in the involution. Of course, this mathemat-ical construction will be justified only if we are capable to obtain HamiltoniansH1,2, that are applicable in some physical models.


4.1. CASE q �=0

Inspired by [14,17] and by Valent example let us substitute

x =a v−1k , z =a0 pvk , k =1,2, a,a0 ∈C, (4.33)

into the generic equation of the (3,4) algebraic curve

(z, x)= z3 + (a1x +a2)z2 + (H1x2 +b1x +b2)z + x4

+H2x3 + c1x2 + c2x + c3 =0, ak,bk, ck ∈C. (4.34)

see [3,4]. Solving the resulting equations with respect to H1,2, one gets the follow-ing Hamilton function

H1 = T + V +(

c2 + ia0b1w2 −a20a1w

22

a0aw2h + 2a0a1w2 − ib1

aw2

)ip1

−gw2(c2 + ia0b1w2 −a20a1w

22)

aa0p2, (4.35)

where geodesic Hamiltonian T and potential V are equal to

T =(

a20(h2w2

2 −3hw2 +3)

a2− ia0a2(hw2 −1)2

a2w2− b2h(hw2 −1)

a2w2+ ic3h2

a0a2w2

)p2

1

+ iga2w2

((2a2

0w32 −2ia0a2w

22 −2b2w2 + 2ic3

a0

)h −3a2

0w22 +2ia0a2w2 +b2

)p1 p2

+ g2(a0b2w2 + ia20a2w

22 −a3

0w32 − ic3)

a2a0w2p2

2

V =− ia2e−iq1

αa0w1w2+ (a0b2w2 + ia2

0a2w22 −a3

0w32 − ic3)αw1eiq1

a0a2w2+ ic1

a0w2.

Here we suppose that h(q2) �=0, g(q2) �=0 and put

w1 = exp(

−∫

h(q2)

g(q2)dq2

), w2 =

∫dq2

g(q2).

The second integral of motion H2 is a cubic polynomial in momenta p1,2, whichhas a more complicated form than (1.2).

The resulting Hamiltonian H1 (4.35) has a natural form, if and only if linear inmomenta terms are equal to zero, i.e. at

2a0a1w2 − ib1 =0, and c2 + ia0b1w2 −a20a1w

22 =0.

So, because w2 �=0, we have to put

a1 =b1 = c2 =0.

If we want to obtain a diagonal metric, then we have to solve the following inte-gral equation

2h(a30w3

2 − ia20a2w

22 −a0b2w2 + ic3)−3a3

0w22 +2ia2

0a2w2 +a0b2 =0, (4.36)


with respect to functions h(q2),w2(q2) and parameters a0,a2,b2, c3. If we want toget a real potential

V = f1(q2) cos(q1)+ f2(q2)

in (4.35), we have to add one more equation to (4.36)

iα2(a30w3

2 − ia20a2w

22 −a0b2w2 + ic3)w

21 +a4 =0 (4.37)

depending on function w1 and parameters a (4.33) and α (4.29). Of course, thereis an additional freedom related to a possible canonical transformation pi → pi +f (qi ) and qi →qi +αi , see [6].

If Q =S2 is a two-dimensional unit sphere with spherical coordinates

q1 =φ, q2 = θ and p1 = pφ, p2 = pθ , (4.38)

then we could get a whole family of natural integrable systems on the sphere usingdifferent functions h(θ), g(θ) labeled by particular values of parameters in (4.36)[14,17]. Note, that the other known system related with (3, 4) algebraic curve is aso-called Hitchin monopole [1].

In the similar manner, we can consider integrable systems associated with thetrigonal curve

(z, x)= z3 + (a1x +a2)z2 + (c1x2 +b1x +b2)z + x4

+H2x3 + H1x2 + c2x + c3 =0, ak,bk, ck ∈C, (4.39)

instead of (4.34). It will be a generalization of the Valent systems at q = 0. Otherpossible generalization is related with another permutations of the integrals ofmotion H1,2 and parameters ak,bk, ck ∈ C in the equation (z, x)= 0 (4.34–4.39).Of course, for other permutations, integrals of motion have a form different from(1.2). Some difficulty here is related to the choice of coordinates, because we can-not directly recognize interesting physical models in ζ -variables.

4.2. THREE-DIMENSIONAL SPHERE

According to [14], there are natural generalizations of matrices � and � (4.28,4.29) on the three-dimensional case. For instance, one of the possible generaliza-tion of � (3.12, 4.28) in three-dimensional case looks like

�=

⎛

⎜⎜⎜⎜⎜⎝

αp21 βp2

1f (q3)

h(q3)F− α f (q3)+βg(q3)

h(q3)p2

1

0 0(

γ∂

∂q3+ g(q3)

h(q3)

)F

0 0 F

⎞

⎟⎟⎟⎟⎟⎠, α, β, γ ∈C,


where element F =(

f (q3)p1 + g(q3)p2 + h(q3)p3

)2depends on three arbitrary

functions f, g, h. The corresponding potential matrix � in (2.5) is a diagonal matrixas in two-dimensional case (3.13, 4.29).

It can allow us to easily obtain new three-dimensional natural integrable sys-tems with higher order integrals of motion. So, different possible generalizations ofthe Goryachev, Chaplygin, Dullin–Matveev systems and other systems from [14,17]associated with the generic trigonal curve in term of physical variables on thethree-dimensional sphere will be discussed in a forthcoming publication.

Acknowledgements

We are greatly indebted referees for several improvements and corrections inducedby their comments.

References

1. Braden, H.W., Enolski, V.Z.: On the tetrahedrally symmetric monopole. Commun.Math. Phys. 299, 255–282 (2010)

2. Braden, H., Enolskii, V.Z., Fedorov, Yu.N.: Dynamics on strata of a trigonal Jacobi-ans in some integrable problems of rigid body, talk on “Finite Dimensional IntegrableSystems in Geometry and Mathematical Physics”, 26–29 July, Jena, Germany (2011)

3. Buchstaber, V.M., Enolskii, V.Z., Leykin, D.V.: Uniformization of Jacobi varieties oftrigonal curves and nonlinear equations. Funct. Anal. Appl. 34, 159–171 (2000)

4. Eilbeck, J.C., Enolski, V.Z., Matsutani, S., Onishi, Y., Previato, E.: Abelian func-tions for trigonal curves of genus three. International Mathematics Research Notices,vol. 2007, rnm 140, p. 38 (2007)

5. Falqui, G., Pedroni, M.: Separation of variables for bi-Hamiltonian systems. Math.Phys. Anal. Geom. 6, 139–179 (2003)

6. Grigoryev, Yu.A., Tsiganov, A.V.: Separation of variables for the generalizedHenon-Heiles system and system with quartic potential. J. Phys. A Math. Theor.44, 255202 (2011)

7. Jacobi, C.G.J.: Vorlesungen uber dynamik. G. Reimer, Berlin (1884)8. Ibort, A., Magri, F., Marmo, G.: Bihamiltonian structures and Stackel separability.

J. Geom. Phys. 33, 210–228 (2000)9. Kiyohara, K.: Two-dimensional geodesic flows having first integrals of higher degree.

Math. Ann. 320, 487–505 (2001)10. Selivanova, E.N.: New examples of integrable conservative systems on S2 and the case

of Goryachev–Chaplygin. Commun. Math. Phys. 207, 641–663 (1999)11. Tsiganov, A.V.: Duality between integrable Stackel systems. J. Phys. A Math. Gen. 32,

7965–7982 (1999)12. Tsiganov, A.V.: The Maupertuis principle and canonical transformations of the

extended phase space. J. Nonlinear Math. Phys. 8, 157–182 (2001)13. Tsiganov, A.V.: On bi-integrable natural hamiltonian systems on Riemannian mani-

folds. J. Nonlinear Math. Phys. 18(2), 245–268 (2011)14. Tsiganov, A.V.: On natural Poisson bivectors on the sphere. J. Phys. A Math. Theor.

44, 105203 (2011)


15. Valent, G.: On a class of integrable systems with a cubic first integral. Commun. Math.Phys. 299, 631–649 (2010)

16. Vanhaecke, P.: Integrable systems and moduli spaces of rank two vector bundles on anon-hyperelliptic genus 3 curve. Annales de l’Institut Fourier 55, 1789–1802 (2005)

17. Vershilov, A.V., Tsiganov, A.V.: On bi-Hamiltonian geometry of some integrable sys-tems on the sphere with cubic integral of motion. J. Phys. A Math. Theor. 42,105203 (2009)

18. Weierstrass, K.: Mathematische Werke I, vol. 1 (1894)

on one integrable system with a cubic first integral

Documents