Two Flows Kowalevski Top as the Full Genus Two Jacobi's Inversion Problem and Sp(4,$\mathbb{R}$) Lie Group Structure

By using the first and the second flows of the Kowalevski top, we can make the Kowalevski top into the two flows Kowalevski top, which has two time variales. Then we show that equations of the two flows Kowalevski top become those of the full genus two Jacobi inversion problem. In addition to the Lax pair for the first flow, we costruct Lax pair for the second flow. Using the first and the second flows, we show that the Lie group structure of these two Lax pairs is Sp(4,$\mathbb{R}$) $\cong$ SO(3,2). Through the two flows Kowalevski top, we can conclude that the Lie group structure of the genus two hyperelliptic function is Sp(4,$\mathbb{R}$) $\cong$ SO(3,2).

We have a dogma that the reason why some special non-linear differential equations have a series of infinitely many solutions is that such non-linear systems have the Lie group structure.
Owing to an addition formula of the Lie group, we obtain a series of infinitely many solutions. As a representation of the addition formula of the Lie group, algebraic functions will emerge, such as trigonometric/elliptic/hyperelliptic functions, as the solutions of such special nonlinear differential equations. A product of Lie group elements is given by an addition of exponential arguments of corresponding Lie algebra, and such multiplication formula is usually called an addition formula of the Lie group. In the KdV case, the Bäcklund formula plays a role of an addition formula to provide new soliton solutions.
Here, we would like to revisit the famous Kowalevski top [25,26]. It is quite surprising that the Kowalevski top was first solved more than one hundred years ago, yet it remains in progress [27][28][29][30][31][32][33][34][35][36][37]. The original first flow (time-dependent) Kowalevski top can be formulated into the special genus two Jacobi's inversion problem in the form where f 5 pXq denotes some fifth-degree polynomial function of X.
In this research, we study the generalized two-flows Kowalevski top by using the second flow of the Kowalevski top [29,30]. We will demonstrate that equations of the two-flows Kowalevski top become those of the full genus two Jacobi's inversion problem. Next, in addition to the Lax pair of the original Kowalevski top [28,31], we will provide the Lax pair for the second flow of the Kowalevski top. Using the Lax pairs of the first and second flows, we will demonstrate that Lax pairs have SO(3,2)-Sp(4,R)/Z 2 Lie group structure. Combining these two results, we conclude that the genus two hyperelliptic function has the SO(3,2)-Sp(4,R)/Z 2 Lie group structure.
2 Two flows Kowalevski top as the full genus two Jacobi's inversion problem

Review of Kowalevski's work
We first review Kowalevski's work [25,26]. A set of equations that describe the rotation of a rigid body around a fixed point under uniform gravitational force with a magnitude of M g is given in the form The coordinates of the center of mass and of the angular velocity are given in a frame attached to the rigid body with the fixed point as the origin, whose axes coincide with the principal axes of inertia. pA, B, Cq represent its principal moments of inertia, while pξ 0 , η 0 , ζ 0 q denotes the corresponding coordinates of the center of mass of the rigid body. Direction cosines pγ 1 , γ 2 , γ 3 q indicate the direction of the axis of rotation, while an angular velocity vector pω 1 , ω 2 , ω 3 q describes the rotation about the axis. The direction cosines are taken between the vertical axis (pointing down) and the three axes of the rigid body frame. This system has six variables.
For the Kowalevski top, we take A " B " 2C, ζ 0 " 0, which provides four conserved quantities. In this case, the center of mass is on the ξη plane. Owing to the fact that A " B, we can rotate the ξ and η axes around ζ axis. Hence, we can put η 0 " 0 without losing generality. Then, the set of equations for the Kowalevski top becomes Here, we have used c 0 "´M gξ 0 {C as a convenient notation. In this Kowalevski's case, we define new variables as These variables satisfy the following equations dq Kowalevski's conserved quantity: pc 0 p γ 1 q 2`p c 0 p γ 2 q 2 " k 2 (2.13) Here, we have introduced the parameters ℓ 1 , ℓ, and k that represent values of each conserved quantity, which were used in the original Kowalevski's work [25]. In the following, we adopt a convenient complex valuable defined as (2.14) We regard ξ and its complex conjugateξ as independent valuables and denote ξ 1 and ξ 2 , respectively. Note that |ξ| 2 " ξ 1 ξ 2 " k 2 . In addition, we introduce other complex variables x 1 and x 2 as We also consider x 1 and x 2 to be independent. In the following, we derive differential equations satisfied by x 1 and x 2 , which will lead to the Jacobi's inversion problem. By using these conserved quantities, ω 3 and γ 3 are solved with variables of Eqs.(2.14) and (2.15) in the form where we used x 2 1 x 2 2 " pω 2 1`ω 2 2 q 2 , x 2 1 ξ 2`x 2 2 ξ 1 " 2pω 2 1`ω 2 2 q 2`2 c 0`p ω 2 1´ω 2 2 qγ 1`2 ω 1 ω 2 γ 2˘. In the right-hand sides of Eqs.(2.16)-(2.18), we have separated the expressions into two parts, i.e., a part which depends on ξ 1 , ξ 2 and others with From an identity ω 2 3ˆc 2 0 γ 2 3 " pc 0 ω 3 γ 3 q 2 , we have where Rpxq " Ax 2`2 Bx`C "´x 4`6 ℓ 1 x 2`4 ℓc 0 x`c 2 0´k 2 , (2.20) To this point, we have adopted six dynamical variables x 1 , x 2 , ξ 1 , ξ 2 , ω 3 , γ 3 so far. Assuming that x 1 and x 2 have been solved, ξ 1 and ξ 2 can be obtained in principle from the relationship and Eq. (2.19). In addition, ω 3 and γ 3 could be obtained from Eqs.(2.16) and (2.18), respectively. Accordingly, we can reduce the number of dynamical variables from six to two, i.e., x 1 and x 2 with four conserved quantities. In the calculation process below, there appears the following combination Rpx 1 qRpx 2 q´px 1´x2 q 2 R 1 px 1 , x 2 q, which gives We note that Rpx 1 , x 1 q " Rpx 1 q and Rpx 2 , x 2 q " Rpx 2 q. Let us consider differential equations for two dynamical variables x 1 and x 2 : (2.29) Then, we have´4 Rpx 1 qˆd Owing to the identity of Eq.(2.19), we can eliminate ξ 1 , ξ 2 by considering the following combination of Eqs.
where we have used Eq.(2.23). Then, we havẽ Next, we introduce Kowalevski variables s 1 , s 2 in the form In terms of the Kowalevski variables, we have a while Eq.(2.34) gives˜d We now assume that the following ϕpsq exists Later, we will explicitly prove that ϕpsq is the third-degree polynomial function of s, which implies that the above assumption is true.
which gives a special genus two Jacobi's inversion problem of the form A general genus two Jacobi's inversion problem is written in the form [38] where f 5 pXq is a fifth-degree polynomial function of X. We will use the term "full" Jacobi's inversion problem for the case where du 1 ‰ 0 and du 2 ‰ 0 and "special" for du 1 " 0 and du 2 ‰ 0. From Eq.(2.44), we have By making the correspondence of The existence of ϕpsq that satisfies Eqs.(2.48) implies that the original Kowalevski top can be expressed in the genus two Jacobi's inversion problem. First, we consider the limit x 2 Ñ x 1 and take the most singular term. In this limit, Rpx 1 , x 2 q Ñ Rpx 1 q, Rpx 2 q Ñ Rpx 1 q. Then, we have This implies that ϕps 2 q " 4s 3 2`( lower order), which is the third-degree polynomial function of s, i.e., ϕpsq " 4s 3`k 2 s 2`k 1 s`k 0 . (2.51) The coefficients k 2 , k 1 , and k 0 are determined by using the relationship (2.52) px 1´x2 q 6ˆ( l.h.s of Eq.(2.52)) represents the polynomial of x 1 and x 2 that involves k 2 , k 1 , and k 0 . By using the definitions of s 1 and s 2 given in Eqs. (2.35), px 1´x2 q 6ˆ( r.h.s of Eq.(2.52)) is proven to be the polynomial of x 1 and x 2 . Hence, from Eq.(2.52), the coefficients k 2 , k 1 , and k 0 are determined. Accordingly, we obtained ϕpsq in the form ϕpsq " 4s 3`6 ℓ 1 s 2´p k 2´c2 0 qs´3 2 ℓ 1 pk 2´c2 0 q´ℓ 2 c 2 0 .

Second flow of the Kowalevski Top
We generalize the original Kowalevski top to two-flows Kowalevski top, and we replace ω i ptq Ñ ω i pr, tq, γ i ptq Ñ γ i pr, tq pi " 1, 2, 3q by introducing another time r, such that x 1 ptq Ñ x 1 pr, tq, x 2 ptq Ñ x 2 pr, tq, s 1 ptq Ñ s 1 pr, tq, s 2 ptq Ñ s 2 pr, tq and d{dt Ñ B{Bt. We call the set of Eqs.(2.1)-(2.6) as the first flow equations. The second flow equations are given in the form [29,30] 2 Under these second flow equations, Eqs.(2.10)-(2.13) are also conserved. Furthermore, the first and second flows are integrable, i.e., the integrability conditions of ω i , γ i pi " 1, 2, 3q are satisfied. For example, the integrability condition B r B t ω 1 " B t B r ω 1 is equivalent to B r pω 2 ω 3 q " B tˆ´c0 γ 2 γ 3`ω3 pγ 1 ω 2´γ2 ω 1 q´2γ 3 ω 1 ω 2´1 c 0 pω 2  It can be considered as the integrability condition by identifying ℘ 22 , ℘ 21 as the genus two hyperelliptic ℘ functions with du 1 : du 2 " 2dr{c 0 : dt. This fact suggests that we have a full Jacobi inversion problem in the form (2.70) The following equations are derived from the sum and difference between Eqs.(2.69) and (2.70);

Lax pairs for two flows Kowalevski top and Sp(4,R) Lie group structure
We usually formulate integrable models with a Lax pair to explicitly demonstrate an integrability of models and systematically obtain various conserved quantities. Let us consider the KdV equation of the form u t´uxxx`6 uu x " 0. From one Lax pair L, B of the form B t L " rB, Ls gives the KdV equation. From another Lax pair L, B, which is known as the AKNS formalism, of the form where λ represents a constant spectral parameter, rB x´L , B t´B s " B t L´B x B´rB, Ls " 0 gives the KdV equation. We construct a Lax pair to examine the Lie group structure of an integrable model. As an infinitesimal Lie group transformation of an integrable model, we formulate a Lax pair using Lie algebra elements with only linear partial differential operators. The Lax pair of Eq.

The bases for the Sp(4,R) Lie algebra
We first construct the bases of the Sp(4,R) Lie algebra by using an almost complex structure J, which is a skew symmetric real matrix with J 2 "´1. For the Sp(4,Rq case, we take the representation J "¨0 Then, the Lie algebra of Sp(4,R) is represented by the spaces of matrices A " pa ij q, p1 ď i, j ď 4q, which satisfy JA`A T J " 0. It is a ten-dimensional representation spanned by the following basis elements

Lax pairs of the first and second flows
The Lax equation of the first flow is given by BL Bt " rB 2 , Ls, (3.4) where the operators L and B 2 have the following form where λ is a constant spectral parameter. For the first flow, we denote the time-evolving operator as B 2 instead of B 1 , because dt is proportional to the second argument du 2 of the Jacobi's inversion relationship of Eq.(2.44). Eq.(3.4) gives Eqs.(2.1)-(2.6). In constructing a Lax pair of L and B 2 , eight Lie algebra elements pI 1`I2`I8`I9 q, pI 1´I2´I8`I9 q, pI 3´I10 q, pI 4´I5 q, pI 4`I5 q, pI 6´I7 q, pI 8´I9 q, I 10 , emerge, i.e., pI 1`I9 q, pI 2`I8 q, pI 6´I7 q, pI 8´I9 q, I 3 , I 4 , I 5 , and I 10 are necessary. Then, in the case of the first flow, we cannot conclude that all ten bases of Sp(4,R) Lie algebra are necessary. Next, we construct the Lax pair of the second flow. The operator L is the same as Eq. (3.5). Assuming that the second flow can be written by the basis of the Sp(4,R) Lie algebra, we rearrange B 1 in the form " a 11 I 3`a12 I 1`a13 I 4`a14 I 6`a21 I 2 a 23 I 7`a24 I 5`a33 I 10`a34 I 8`a43 I 9 . (3.10) The Lax equation for the second flow is given by To ensure that this Lax equation gives Eqs.(2.54)-(2.59), ten independent linear relationships for ten variables a 11 , a 12 , a 13 , a 14 , a 21 , a 23 , a 24 , a 33 , a 34 , and a 43 must be satisfied. These ten equations to be satisfied are provided in Appendix A. However, the expressions of the solutions are too long, and the entire expressions cannot be presented in this paper; hence, we present only the first few terms in Appendix A. To achieve our , it is not necessary to know the explicit form of B 1 ; however it is sufficient that there really exist solutions. By using these solutions a ij 's, we have confirmed that the Lax equation B r L " rB 1 , Ls is completely satisfied by using the second flow equations Eqs.(2.54)-(2.59). As a 23 "´1, a 24 "´2, we have a 24 " 2a 23 ; however other a ij 's are independent, such that we need nine bases I 1 , I 2 , I 3 , I 4 , I 6 , I 8 , I 9 , I 10 , pI 7`2 I 5 q, (3.12) for the Lax pair of the second flow. However, for the first flow, I 5 is required. Then I 7 itself becomes necessary in the combination of both flows. This fact implies that all of the ten Sp(4,R)/Z 2 -SO(3,2) Lie algebra bases are necessary and sufficient to construct Lax pairs of the first and second flows for two-flows Kowalevski top. It is important that coefficients of all ten Sp(4,R)/Z 2 Lie algebra bases are independent. If some coefficients of the Lie algebra bases are not independent, there is a possibility that the structure of the Lie algebra of two flow Kowalevski top reduces to some Lie subalgebra of the Sp(4,R)/Z 2 Lie algebra. In the Lax formalism, it is quite non-trivial and is quite difficult to find the necessary and sufficient Lie algebra to obtain the two-flows Kowalevski top. In such a situation, we say that the two-flows Kowalevski top has the Lie group structure.
On one hand, we demonstrated that the two-flows Kowalevski top is equivalent to the full genus two Jacobi's inversion problem in Section 2. On the other hand, we proved that the Lie group structure of Lax pairs of two-flows Kowalevski top is Sp(4,R)/Z 2 -SO (3,2) in Section 3. Combining these results, we conclude that the genus two hyperelliptic function has the Sp(4,R)/Z 2 -SO(3,2) structure.

Summary and Discussions
First, we reviewed the Kowalevski top to explain the formulation and notation, as it is quite prevalent, yet was first introduced more than one hundred years ago. The equations of the original Kowalevski top, or the first flow, can be formulated into the special genus two Jacobi's inversion problem. In addition to the first flow, we have adopted the second flow, i.e., other equations of the Kowalevski top by introducing another time r. The first and second flows satisfy the integrability condition. In addition, we demonstrated that the two-flows Kowalevski top is formulated into the full genus two Jacobi's inversion problem.
Next, in addition to the Lax pair of the first flow for the Kowalevski top, we constructed the Lax pair of the second flow for the Kowalevski top. Then, we deduced the Sp(4,R)/Z 2 -SO(3,2) Lie group structure of Lax pairs of the first and second flows. Combining these results, using the two-flows Kowalevski top, we infer that the genus two hyperelliptic function, which is the solution of the full genus two Jacobi's inversion problem has the Sp(4,R)/Z 2 -SO(3,2) Lie group structure.