2.1. Basic Notions of Heavy Rigid Body Fixed at a Point.

A three-dimensional rigid body is a system of material points in $ \mathbb{R}^3$ such that the distance between each two points is a constant function of time. Important case of motion is when rigid body moves with fixed point $ O$. Then the configuration space is the Lie group $ SO(3)$. In order to describe the motion, it is usual to introduce two Euclidian frames associated to the system: the first one $ Oxyz$ is fixed in the space, and the second, moving, $ OXYZ$ is fixed in the body. The capital letters will denote elements of the moving reference frame, while the lowercase letters will denote elements of the fixed reference frame. Let $ B(t)\in SO(3)$ is an orthogonal matrix which maps $ OXYZ$ to $ Oxyz$. The radius vector $ \vvec{Q}$ of the arbitrary point in the moving coordinate system maps to the radius vector in the fixed frame $ \vvec{q}(t)=B(t)\vvec{Q}$. The velocity of that point in the fixed reference frame is given by \begin{eqnarray*} \vvec{v}(t)=\dot{\vvec{q}}(t)=\dot{B}(t)\vvec{Q}=\dot{B}(t)B^{-1}(t)\vvec{q}(t)=\omega(t)\vvec{q}(t), \end{eqnarray*} where $ \omega(t)=\dot{B}B^{-1}$. The matrix $ \omega$ is an skew-symmetric matrix. Using the isomorphism of $ (\mathbb{R}^3, \times)$, where $ \times$ is the usual vector product, and $ (so(3), [\,,])$, given by

In the moving reference frame, $ \vvec{V}(t)=B(t)^{-1}\vvec{v}(t)$, so $ \vvec{V}(t)=\vvec{\Omega}(t)\times\vvec{Q}$, where $ \vvec{\Omega}(t)$ is the angular velocity in the moving reference frame and corresponds to the skew-symmetric matrix $ \Omega(t)=B^{-1}(t)\dot{B}(t)$.

Here one concludes that it is natural to consider the angular velocity as a skew-symmetric matrix. The element $ \omega_{12}$ corresponds to the rotation in the plane determined by the first two axes $ Ox$ and $ Oy$, and similarly for the other elements. In the three-dimensional case we have a natural correspondence given above, and one can consider the angular velocity as a vector. But, in higher-dimensional cases, generally speaking, such a correspondence does not exist. We will see later how in dimension four, using isomorphism between $ so(4)$ and $ so(3)\times so(3)$ two vectors in the threedimensional space are joined to an $ 4\times 4$ skew-symmetric matrix. The moment of inertia with respect to the axis $ u$, defined with the unit vector $ \vvec{u}$ through a fixed point $ O$ is : \begin{eqnarray*} I(u)=\int_B d^2dm=\int_B \langle\vvec{u}\times\vvec{Q},\vvec{u}\times\vvec{Q}\rangle dm=\int_B\Big\langle \vvec{Q}\times (\vvec{u}\times\vvec{Q}), \vvec{u}\Big\rangle dm=\langle I\vvec{u},\vvec{u}\rangle, \end{eqnarray*} where $ d$ is the distance between corresponding point and axis $ u$, $ I$ is inertia operator with respect to the point $ O$ defined with \begin{eqnarray*} I\vvec u=\int_B \vvec{Q}\times(\vvec{u}\times\vvec{Q})dm, \end{eqnarray*} and integrations goes over the body $ B$. The diagonal elements $ I_1, I_2, I_3$ are called the principal moments of inertia , with respect to the principal axes of inertia . The ellipsoid $ \langle I\Omega,\Omega\rangle=1$ is the inertia ellipsoid of the body at the point $ O$. In the principal coordinates its equation is: \begin{eqnarray*} I_1\Omega_1^2+I_2\Omega_2^2+I_3\Omega_3^2=1. \end{eqnarray*} The kinetic energy of the body is given by: \begin{eqnarray*} T&=&\frac12\int_B V^2dm=\frac12\int_B \langle\vvec{\Omega}\times\vvec{Q}, \vvec{\Omega}\times\vvec{Q}\rangle dm\\ &=&\frac12\langle\vvec{\Omega}, \int_B \vvec{Q}\times(\vvec{\Omega}\times\vvec{Q}) dm\rangle=\frac12\langle I\vvec{\Omega}, \vvec{\Omega}\rangle \end{eqnarray*} Similarly, for the angular momentum $ \vvec{M}$ with respect to the point $ O$, we have: \begin{eqnarray*} \vvec{M}=\int_B \vvec{Q}\times \vvec{V}dm=\int_B \vvec{Q}\times(\vvec{\Omega}\times\vvec{Q})dm=I\vvec{\Omega}. \end{eqnarray*}

We consider a motion of a heavy rigid body fixed at a point. Let us denote by $ \vvec{\chi}$ the radius vector of the center of masses of the body multiplied with the mass $ m$ of the body and the gravitational acceleration $ g$. By $ \vvec{\Gamma}$ we denote the unit vertical vector.

The motion in the moving reference frame is described by the Euler- Poisson equations ([Leimanis1965], [Whittaker1952], [Golubev1953], [Borisov and Mamaev2001]):

Since the equations preserve the standard measure, by the Jacobi theorem (see for example [Golubev1953], [Arnold et al.2009]) for integrability in quadratures one needs one more additional functionally independent first integral.

On the other hand, the Eq. (2.2) are Hamiltonian on the Lie algebra $ e(3)$ with the standard Lie-Poisson structure:

Thus a natural problem arises: for which values of the parameters $ I_1, I_2, I_3$, $ X_0, Y_0, Z_0$, the Eq. (2.2) admit the fourth functionally independent first integral?

2.2. Integrable Cases.

The existence of an additional independent fourth integral gives strong restrictions on the moments of inertia and the vector $ \vvec{\chi}$. Such an integral exists in the three cases ([Euler1765], [Lagrange1788], [Kowalevski1889]) (see also [Golubev1953], [Leimanis1965], [Whittaker1952], [Kozlov1995], [Borisov and Mamaev2001], [Arkhangel'skiy1977]):

• Euler case (1751): $ X_0=Y_0=Z_0=0$. The additional integral is $ F_4=\langle M, M\rangle$.
• Lagrange case ([Lagrange1788]): $ I_1=I_2$, $ \vvec{\chi}=(0,0,Z_0)$. The additional integral is $ F_4=M_3$.
• Kovalewski case ([Kowalevski1889]): $ I_1=I_2=2I_3$, $ \vvec{\chi}=(X_0,0,0)$. The additional integral is $ F_4=(\Omega_1^2-\Omega_2^2+\frac{X_0}{I_3}\Gamma_1)^2+(2\Omega_1\Omega_2+\frac{X_0}{I_3}\Gamma_2)^2$.

There are also cases that admit a fourth first integral only for a fixed value of one of the integrals. If the Casimir function $ F_1=0$, then we have

• Goryachev-Chaplygin case (1900): $ I_1=I_2=4I_3,\ \vvec\chi=(X_0,0,0)$. The additional integral is $ F_4=M_3(M_1^2+M_2^2)+2M_1\Gamma_3$;

Following the Kowalevski paper ([Kowalevski1889]), a natural problem arises: to find all cases of the Euler-Poisson equations that admit an additional fourth first integral. Using the results of Liouville, in [Husson1906] Husson proved that an additional algebraic integral exists only in the Euler, Lagrange and Kovalewski cases. Simplified proofs of Liouville's and Husson's results were presented by Dokshevich (see [Dokshevich1974]). On the other hand, Poincaré considered a more general problem of the existence of an analytical first integral of the canonical systems. Using the method of a small parameter, he developed a tool for proving nonintegrability of a perturbation of an integrable Hamiltonian system. However, Poincaré observed that his method cannot be applied to the Euler-Poisson equations. In 1970's Kozlov in [Kozlov1975] (see also [Kozlov1980], [Arkhangel'skiy1977]) modified the Poincaré results and proved that a nonsymmetric rigid body does not admit an additional analytical integral except in the Euler case. The case of a symmetric rigid body is even more complicated. The nonexistence of an additional (complex or real-valued) analytical or meromorphic integral except in the three classical cases was finally proved in the papers of [Kozlov and Treschev1985], [Kozlov and Treschev1986], [Ziglin1997]. Ziglin also proved that having the value of $ F_1$ fixed to be zero, an additional meromorphic integral exists only in one extra case--the Goryachev- Chaplygin case.

2.3. Definition of the Hess-Appel'rot System.

Beside the completely integrable cases, there are classically well-known systems which possess an invariant relation instead of an additional first integral. A list of such systems can be found, for example in [Gorr et al.1978].

Some of these cases where obtained using new forms of the Euler- Poisson equations and a method of constructing invariant relations given by Kharlamov (for details see [Kharlamov1965], [Kharlamov1974a], [Kharlamov1974b], [Kharlamov and Kovalev1997], [Gorr et al.1978], [Gashenenko et al.2012]). We will focus on the Hess-Appel'rot case.

It is well known that Kowalevski, in her above mentioned celebrated 1889 paper [Kowalevski1889], started with a careful analysis of the solutions of the Euler and the Lagrange case of rigid-body motion, and formulated a problem to describe the parameters $ (I_1, I_2, I_3, X_0, Y_0, Z_0)$, for which the Euler-Poisson equations have a general solution in a form of a uniform (single-valued) function having moving poles as the only possible singularities.

Then, some necessary conditions were formulated in [Kowalevski1889] and a new case was discovered, now known as the Kowalevski case, as a unique possible beside the cases of Euler and Lagrange. However, considering the situation where all the momenta of inertia are different, Kowalevski came to a relation analogue to the following: \begin{eqnarray*} X_0\sqrt{I_1(I_2-I_3)}+Y_0\sqrt{I_2(I_3-I_1)}+Z_0\sqrt{I_3(I_1-I_2)}=0, \end{eqnarray*} and concluded that the relation $ X_0=Y_0=Z_0$ follows, leading to the Euler case.

But, it was Appel'rot (see [Appel'rot1892], [Appel'rot1894]) who noticed in the beginning of 1890's, that the last relation admitted one more case, not mentioned by Kowalevski:

The system that satisfies the conditions (2.5) was considered also by Hess, even before Appel'rot, in 1890. [Hess1890] found that if the inertia momenta and the radius vector of the center of masses satisfy the conditions (2.5) , then the surface

2.4. A Lax Representation for the Classical Hess-Appel'rot System: An Algebro-Geometric Integration Procedure.

A Lax representation for the classical Hess-Appel'rot system, with an algebro-geometric integration procedure was presented in [Dragovi'c and Gaji'c2001]. The classical integration procedure leads to an elliptic function and an additional Riccati equation (see [Golubev1953]). In [Dragovi'c and Gaji'c2001] an algebro-geometric integration procedure was presented with the same properties.

Using isomorphism (2.1) , Eq. (2.2) can be written in the matrix form: \begin{eqnarray*} \dot{M}&=&[M,\Omega]+[\Gamma,\chi]\\ \dot{\Gamma}&=&[\Gamma,\Omega], \end{eqnarray*} where the skew-symmetric matrices represent vectors denoted by the same letter.

We have the following:

The spectral curve is defined by: \begin{eqnarray*} \mathcal{C}: \ \ p(\mu,\lambda):=\det(L(\lambda)-\mu E)=0, \end{eqnarray*} is: \begin{eqnarray*} \mathcal{C}: \ \ -\mu(\mu^2-\omega^2+2\Delta\Delta^*)=0 \end{eqnarray*} where

The coefficients of the spectral polynomial are integrals of motion. If one rewrites the equation of the spectral curve in the form: \begin{eqnarray*} p(\mu, \lambda)=-\mu(\mu^2+A\lambda^4+B\lambda^3+D\lambda^2+E\lambda+F)=0, \end{eqnarray*} one gets: \begin{eqnarray*} A&=&I_2^2(X_0^2+Z_0^2),\\ B&=&2I_2(M_1X_0+M_3Z_0)(=0),\\ D&=&M_1^2+M_2^2+M_3^2+ 2I_2(X_0\Gamma_1+Z_0\Gamma_3),\\ E&=&2(M_1\Gamma_1+M_2\Gamma_2+M_3\Gamma_3),\\ F&=&\Gamma_1^2+\Gamma_2^2+\Gamma_3^2(=1). \end{eqnarray*} Thus, the L-A pair (2.7) gives three first integrals and one invariant relation.

Now, we review some basic steps in the algebro-geometric integration procedure from [Dragovi'c and Gaji'c2001]. Let $ (f_1, f_2, f_3)^T$ denote an eigenvector of the matrix $ L(\lambda)$, which corresponds to the eigenvalue $ \mu$. Fix the normalizing condition $ f_1=1$. Then one can prove:

We are going to analyze the converse problem. Suppose the evolution in time of the point $ \nu$ is known. For reconstructing the matrix $ L(\lambda)$, one needs $ x=|x|e^{i\arg x},\ y=|y|e^{i\arg y}$ as functions of time.

Thus, in order to determine $ L(\lambda)$ as a function of time, one needs to find the evolution of the point $ \nu$ and $ \arg x$ as a function of time. In [Dragovi'c and Gaji'c2001] the following two theorems are proved:

Denote by $ \phi_x=\arg x$, and $ u=\tan \frac{\phi_x}{2}$.

In recent years some other methods have been applied as well to study the Hess-Appel'rot system (see [Borisov and Mamaev2003], [Lubowiecki and &Ddot;ołądek2012a], [Lubowiecki and &Ddot;ołądek2012b], [Belyaev2015], [Simić2000]).

2.5. Zhukovski's Geometric Interpretation.

In [Zhukovski1894] Zhukovski gave a geometric interpretation of the Hess-Appel'rot conditions. Denote $ J_i=1/I_i$. Consider the so-called gyroscopic inertia ellipsoid: \begin{eqnarray*} \frac {M_1^2}{J_1}+\frac {M_2^2}{J_2}+\frac {M_3^2}{J_3}=1, \end{eqnarray*} and the plane containing the middle axis and intersecting the ellipsoid at a circle. Denote by $ l$ the normal to the plane, which passes through the fixed point $ O$. Then the condition (2.5) means that the center of masses lies on the line $ l$.

If we choose a basis of moving frame such that the third axis is $ l$, the second one is directed along the middle axis of the ellipsoid, and the first one is chosen according to the orientation of the orthogonal frame, then (see [Borisov and Mamaev2001]), the invariant relation (2.6) becomes \begin{eqnarray*} F_4=M_3=0, \end{eqnarray*} the matrix $ J$ obtains the form: \begin{eqnarray*} J=\left(\begin{matrix} J_1&0&J_{13}\\ 0&J_1&0\\ J_{13}&0&J_3 \end{matrix}\right), \end{eqnarray*} and $ {\chi}=(0, 0, Z_0)$.

One can see here that the Hess-Appel'rot system can be regarded as a perturbation of the Lagrange top. In new coordinates the Hamiltonian of the Hess-Appel'rot system becomes \begin{eqnarray*} H_{HA}=\frac{1}{2}\left(J_1(M_1^2+M_2^2)+J_3M_3^2\right)+Z_0\Gamma_3+J_{13}M_1M_3=H_L+J_{13}M_1M_3 \end{eqnarray*}

This serves as a motivation for a definition of higher-dimensional Hess- Appel'rot systems in [Dragovi'c and Gaji'c2006], which will be presented in Sect. 4.6.

3.1. Discriminantly Separable Polynomials.

We will start from the equation of a pencil of conics $ \mathcal F(w, x_1, x_2)=0, $ where $ w$, $ x_1$ and $ x_2$ are the pencil parameter and the Darboux coordinates respectively. We recall some of the details: given two conics $ C_1$ and $ C_2$ in general position by their tangential equations

Given the projective plane with the standard coordinates $ (z_1:z_2:z_3)$, we rationally parameterize the conic $ C_2$ by $ (1,\ell,\ell^2)$ as above. The tangent line to the conic $ C_2$ through a point of the conic with the parameter $ \ell_0$ is given by the equation \begin{eqnarray*} t_{C_2}(\ell_0): z_1\ell_0^2-2z_2\ell_0+z_3=0. \end{eqnarray*}

For a given point P outside the conic in the plane with coordinates $ P = (\hat{z}_1,\hat{z}_2,\hat{z}_3)$, there are two corresponding solutions $ x_1$ and $ x_2$ of the equation quadratic in $ \ell$ \begin{eqnarray*} \hat z_1\ell^2 - 2\hat z_2\ell + \hat z_3=0. \end{eqnarray*} Each of the solutions corresponds to a tangent to the conic $ C_2$ from the point $ P$. We will use the pair $ (x_1, x_2)$ as the Darboux coordinates (see [Darboux1917]) of the point $ P$. One finds immediately the converse formulae \begin{eqnarray*} \hat z_1=1,\quad \hat z_2= \frac{x_1+x_2}{2},\quad \hat z_3 = x_1 x_2. \end{eqnarray*}

Changing the variables in the polynomial $ F$ from the projective coordinates $ (z_1:z_2:z_3)$ to the Darboux coordinates, we rewrite its equation $ F$ in the form \begin{eqnarray*} F(s, x_1, x_2)=L(x_1,x_2)s^2 + K(x_1,x_2)s + H(x_1,x_2). \end{eqnarray*} The key algebraic property of the pencil equation written in this form, as a quadratic equation in each of three variables $ s, x_1, x_2$ is: all three of its discriminants are expressed as products of two polynomials in one variable each : \begin{eqnarray*} \mathcal D_s( F)(x_1,x_2)=P(x_1)P(x_2),\quad \mathcal D_{x_i}( F)(s,x_j)=J(s)P(x_j),\quad i,j={1,2}, \end{eqnarray*} where J and P are polynomials of degree $ 3$ and $ 4$ respectively, and the elliptic curves \begin{eqnarray*} \Gamma_1: y^2=P(x), \quad \Gamma_2: y^2=J(s) \end{eqnarray*} are isomorphic (see Proposition 1 of [Dragović2010]).

As a geometric interpretation of $ F(s, x_1, x_2)=0$ we may say that the point $ P$ in the plane, with the Darboux coordinates with respect to $ C_2$ equal to $ (x_1, x_2)$ belongs to two conics of the pencil, with the pencil parameters equal to $ s_1$ and $ s_2$, such that \begin{eqnarray*} F(s_i, x_1, x_2)=0, \quad i=1, 2. \end{eqnarray*}

Now we recall a general definition of the discriminantly separable polynomials. With $ \mathcal{P}_m^n$ denote the set of all polynomials of $ m$ variables of degree $ n$ in each variable.

3.2. Two-Valued Groups.

The structure of formal (local) $ n$-valued groups was introduced by Buchstaber and Novikov ([Buchstaber and Novikov1971]) in their study of characteristic classes of vector bundles. It has been studied further by Buchstaber and his collaborators since then (see [Buchstaber2006] and references therein).

Following [Buchstaber2006], we give the definition of an $ n$-valued group on $ X$ as a map: \begin{eqnarray*} &m:\, X\times X \rightarrow (X)^n\\ &m(x,y)=x*y=[z_1,\dots, z_n], \end{eqnarray*} where $ (X)^n$ denotes the symmetric $ n$-th power of $ X$ and $ z_i$ coordinates therein. Associativity is the condition of equality of two $ n^2$-sets \begin{eqnarray*} &[x*(y*z)_1,\dots, x*(y*z)_n]\\ &[(x*y)_1*z,\dots, (x*y)_n*z] \end{eqnarray*} for all triplets $ (x,y,z)\in X^3$.

An element $ e\in X$ is a unit if \begin{eqnarray*} e*x=x*e=[x,\dots,x], \end{eqnarray*} for all $ x\in X$.

A map $ \inv: X\rightarrow X$ is an inverse if it satisfies \begin{eqnarray*} e\in \inv(x)*x, \quad e\in x*\inv(x), \end{eqnarray*} for all $ x\in X$.

Following Buchstaber, we say that $ m$ defines an $ n$-valued group structure $ (X, m, e, \inv)$ if it is associative, with a unit and an inverse.

An $ n$-valued group $ X$ acts on the set $ Y$ if there is a mapping \begin{eqnarray*} &\phi:\, X\times Y \rightarrow (Y)^n\\ &\phi (x,y)=x\circ y, \end{eqnarray*} such that the two $ n^2$-multisubsets of $ Y$ \begin{eqnarray*} x_1\circ (x_2\circ y) \quad (x_1*x_2)\circ y \end{eqnarray*} are equal for all $ x_1, x_2\in X, y\in Y$. It is additionally required that \begin{eqnarray*} e\circ y=[y,\dots, y] \end{eqnarray*} for all $ y\in Y$.

In the sequel, we are going to show that there is another 2-valued group and its action on $ \mathbb {CP}^1$ which is even more closely related to the Euler-Chasles correspondence and to the Great Poncelet Theorem (see [Dragović and Radnović2011]), and which is at the same time intimately related to the Kowalevski fundamental equation and to the Kowalevski change of variables.

However, we will start our approach with a simple example. The Simplest Case: 2-Valued Group $ p_2$

Among the basic examples of multivalued groups, there are $ n$- valued additive group structures on $ \mathbb C$. For $ n=2$, this is a two-valued group $ p_2$ defined by the relation

The product $ x *_2 y$ corresponds to the roots in $ z$ of the polynomial equation \begin{eqnarray*} p_2(z, x, y)=0, \end{eqnarray*} where \begin{eqnarray*} p_2(z, x, y)= (x+y+z)^2-4(xy+yz+zx). \end{eqnarray*} Our starting point in this section is the following

The polynomial $ p_2$ as a discriminantly separable, generates a case of generalized Kowalevski system of differential equations from [Dragović2010].

3.3. 2-Valued Group Structure on $ \mathbb{CP}^1$ and the Kowalevski Fundamental Equation.

Now we pass to the general case. We are going to show that the general pencil equation represents an action of a two valued group structure. Recognition of this structure enables us to give to 'the mysterious Kowalevski change of variables' (see [Audin1996] for the wording "mysterious") a final algebro-geometric expression and explanation, developing further the ideas of Weil and Jurdjevic (see [Weil1983], [Jurdjevic1999a], [Jurdjevic1999b]). Amazingly, the associativity condition for this action from geometric point of view is nothing else than the Great Poncelet Theorem for a triangle.

As we have already mentioned, the general pencil equation \begin{eqnarray*} F(s,x_1, x_2)=0 \end{eqnarray*} is connected with two isomorphic elliptic curves \begin{eqnarray*} \tilde\Gamma_1: y^2&=P(x)\\ \tilde\Gamma_2: t^2&=J(s) \end{eqnarray*} where the polynomials $ P, J$ of degree four and three respectively. Suppose that the cubic one $ \tilde\Gamma_2$ is rewritten in the canonical form \begin{eqnarray*} \tilde\Gamma_2: t^2=J'(s)=4s^3-g_2s-g_3. \end{eqnarray*} Moreover, denote by $ \psi:\, \tilde\Gamma_2\rightarrow \tilde\Gamma_1$ a birational morphism between the curves induced by a fractional-linear transformation $ \hat \psi$ which maps three zeros of $ J'$ and $ \infty$ to the four zeros of the polynomial $ P$.

The curve $ \tilde\Gamma_2$ as a cubic curve has the group structure. Together with its subgroup $ \mathbb {Z}_2$ it defines the standard two-valued group structure of coset type on $ \mathbb {CP}^1$ (see [Buchstaber1990]):

For the proof see [Dragović2010].

3.4. Fundamental Steps in the Kowalevski Integration Procedure.

Let us recall briefly that the Kowalevski top [Kowalevski1889] is a heavy top rotating about a fixed point, under the conditions $ I_1=I_2=2I_3,\,I_3=1$, $ Y_0=Z_0=0$ (see Sect. 2.1). Denote with $ c=mgX_0$, ($ m$ is the mass of the top), and with $ (p,q,r)$ the vector of angular velocity $ \vvec\Omega$. Then the equations of motion take the following form, see [Kowalevski1889], [Golubev1953]:

The system (3.4) has three well known first integrals of motion and a fourth first integral discovered by Kowalevski

After the change of variables

From (3.8) , Kowalevski gets

After a few transformations, (3.9) can be written in the form

\begin{eqnarray*} \mathcal D_{x_1}(Q)(s,x_2)=-8J(s)P(x_2),\,\mathcal D_{x_2}(Q)(s,x_1)=-8J(s)P(x_1) \end{eqnarray*} with \begin{eqnarray*} J(s)=s^3+3l_1s^2+s(c^2-k^2)+3l_1(c^2-k^2)-2l^2c^2. \end{eqnarray*} The equations of motion (3.4) can be rewritten in new variables $ (x_1, x_2, e_1, e_2, r, \Gamma_3)$ in the form:

Further integration procedure is described in [Kowalevski1889], and in [Dragović and Kukić2014a].

We get the following

The Kowalevski case corresponds to the general case under the restrictions $ a_1=0 \quad a_5=0 \quad a_0=-2.$ The last of these three relations is just normalization condition, provided $ a_0\ne 0$. The Kowalevski parameters $ l_1, l, c$ are calculated by the formulae \begin{eqnarray*} l_1=\frac{a_2}{3} \quad l=\pm\frac{1}{2}\sqrt{-a_4+\sqrt{a_4+4a_3^2}} \quad c=\mp\frac{a_3}{\sqrt{-a_4+\sqrt{a_4+4a_3^2}}} \end{eqnarray*} provided that $ l$ and $ c$ are requested to be real.

Let us mention that Kowalevski in ([Kowalevski1889]), instead the relation (3.11) , used the equivalent one, where the equivalence is obtained by putting $ w=2s-l_1$.

The Kowalevski change of variables is the following consequence of the discriminant separability property of the polynomial $ F=Q$:

The Kowalevski change of variables (see Eq. (3.15) ) is infinitesimal of the correspondence which maps a pair of points $ (M_1,M_2)$ to a pair of points $ (S_1, S_2)$. Both pairs belong to a $ {\mathbb{P}}^1$ as a factor of the appropriate elliptic curve. In our approach, there is a geometric view to this mapping as the correspondence which maps two tangents to the conic $ C$ to the pair of conics from the pencil which contain the intersection point of the two lines .

If we apply fractional-linear transformations to transform the curve $ \tilde\Gamma_1$ into the curve $ \tilde\Gamma_2$, then the above correspondence is nothing else then the two-valued group operation $ *_c$ on $ (\tilde\Gamma_2, \mathbb Z_2)$.

Now, the K"otter trick (see [Kotter1893], [Dragović2010]) can be applied to the following commutative diagram.

Proposition 3.1.

([Dragović2010]) The Kowalevski integration procedure may be codded in the following commutative diagram:

The mappings are defined as follows \begin{eqnarray*} &&i_{\tilde\Gamma_1}:\,x\mapsto (x,\sqrt{P(x)})\\ &&m:\,(x,y) \mapsto x\cdot y\\ &&i_a:\, x \mapsto (x,1)\\ &&p_1:\, (x, y)\mapsto x \\ &&m_c:\, (x, y)\mapsto x*_c y\\ &&\tau_c:\, x\mapsto (\sqrt{x},-\sqrt{x})\\ &&\varphi_1:\, (x_1,x_2, e_1, e_2)\mapsto \sqrt{e_1}\frac{\sqrt{P(x_2)}}{x_1-x_2}\\ &&\varphi_2:\, (x_1,x_2, e_1, e_2)\mapsto \sqrt{e_2}\frac{\sqrt{P(x_1)}}{x_1-x_2}\\ &&f:\, ((s_1,s_2,1),(k,-k))\mapsto [(\gamma^{-1}(s_1)+k)(\gamma^{-1}(s_2)-k), (\gamma^{-1}(s_2)+k)(\gamma^{-1}(s_1)-k)] \end{eqnarray*}

From the Proposition 3.1 we see that the two-valued group plays an important role in the Kowalevski system and its generalizations.

3.5. Systems of the Kowalevski Type: Definition.

Following [Dragović and Kukić2011], [Dragović and Kukić2014a],[Dragović and Kukić2014b], we are going to present a class of dynamical systems, which generalizes the Kowalevski top. Instead of the Kowalevski fundamental equation (see formula (3.11) ), the starting point here is an arbitrary discriminantly separable polynomial of degree two in each of three variables.

Given a discriminantly separable polynomial of the second degree in each of three variables

Suppose additionally, that the first integrals of the initial system reduce to a relation

The equations for $ \dot r$ and $ \dot \Gamma_3$ are not specified for the moment and $ m$ is a function of system's variables.

If a system satisfies the above assumptions we will call it a system of the Kowalevski type . As it has been pointed out in the previous subsection, see formulae (3.8) , (3.11) , (3.12) , (3.13) , the Kowalevski top is an example of the systems of the Kowalevski type.

The following theorem is quite general, and concerns all the systems of the Kowalevski type. It explains in full a subtle mechanism of a quite miraculous jump in genus, from one to two, in integration procedure, which has been observed in the Kowalevski top, and now it is going to be established as a characteristic property of the whole new class of systems.

The last Theorem basically formalizes the original considerations of Kowalevski, in a slightly more general context of the discriminantly separable polynomials. A proof is presented in [Dragović and Kukić2014b].

In the following subsections we present the Sokolov system given in [Sokolov2002] as an example of systems of the Kowalevski type, and one more recent example of the systems of the Kowalevski type.

3.6. An Example of Systems of the Kowalevski Type.

Consider the Hamiltonian (see [Sokolov2002], [Sokolov and Tsiganov2001])

The Lie-Poisson brackets (3.21) have two well known Casimir functions \begin{eqnarray*} &&\gamma_1^2+\gamma_2^2+\gamma_3^2=a,\\ &&\gamma_1M_1+\gamma_2M_2+\gamma_3M3=b. \end{eqnarray*}

Following [Komarov et al.2003] and [Kowalevski1889] we introduce new variables \begin{eqnarray*} z_1=M_1+iM_2, \quad z_2=M_1-iM_2 \end{eqnarray*} and \begin{eqnarray*} e_1&=z_1^2-2c_1(\gamma_1+i\gamma_2)-c_2^2 a-c_2(2\gamma_2 M_3-2\gamma_3 M_2+2i (\gamma_3 M_1-\gamma_1 M_3)),\\ e_2&=z_2^2-2c_1(\gamma_1-i\gamma_2)-c_2^2 a-c_2(2\gamma_2 M_3-2\gamma_3 M_2+2i (\gamma_1 M_3-\gamma_3 M_1)). \end{eqnarray*} The second integral of motion for system (3.20) then may be written as

In [Komarov et al.2003] the biquadratic form and the separated variables are defined as the next step:

\begin{eqnarray*} \dot{z_1}=-2M_3(M_1-iM_2)+2c_2(\gamma_1 M_2-\gamma_2 M_1)+2c_1\gamma_3 \end{eqnarray*} and \begin{eqnarray*} \dot{z_2}=-2M_3(M_1+iM_2)+2c_2(\gamma_1 M_2-\gamma_2 M_1)+2c_1\gamma_3 \end{eqnarray*} one can prove that \begin{eqnarray*} \dot{z_1}\cdot \dot{z_2}=-\left(F(z_1,z_2)+(H+c_2^2a(z_1-z_2)^2)\right) \end{eqnarray*} where $ F(z_1,z_2)$ is given by (3.25) . After equating the square of $ \dot{z_1}\dot{z_2}$ from previous relation and $ \dot{z_1}^2 \cdot \dot{z_2}^2$ with $ \dot{z_i}^2$ given by (3.23) we get

3.7. Another Example of an Integrable System of the Kowalevski Type.

Now, we are going to present one more example of a system of the Kowalevski type. Let us consider the next system of differential equations:

After a change of variables \begin{eqnarray*} x_1 &=& p + q,\quad e_1 = x_1^2 + \gamma_1 + \gamma_2,\\ x_2 &=& p - q, \quad e_2 = x_2^2 + \gamma_1 - \gamma_2, \end{eqnarray*} the system (3.30) becomes

The first integrals of the system (3.31) can be presented in the form

From the integrals (3.32) we get a relation of the form (3.19)

Without loss of generality, we can assume $ h=0$ (this can be achieved by a simple linear change of variables $ x_i \mapsto x_i-h/6,\quad s \mapsto s-h/6$), thus we can use directly the Weierstrass $ \wp$ function. Following the procedure described in Theorem 3.4 we get

Finally, we get

3.8. Another Class of Systems of the Kowalevski Type.

In this section we will consider another class of systems of Kowalevski type. We consider a situation analogue to that from the beginning of the Sect. 3.5. The only difference is that the systems we are going to consider now, reduce to (3.17) , where

3.9. A Deformation of the Kowalevski Top.

In this Section we are going to derive the explicit solutions in genus two theta-functions of the Jurdjevic elasticae [Jurdjevic1999a] and for similar systems ([Komarov1981], [Komarov and Kuznetsov1990]). First, we show that we can get the elasticae from the Kowalevski top by using the simplest gauge transformations of the discriminantly separable polynomials.

Consider a discriminantly separable polynomial \begin{eqnarray*} \mathcal{F}(x_1,x_2,s):=s^2A+sB+C \end{eqnarray*} where

A simple affine gauge transformation $ s\mapsto t + \alpha $ transforms $ \mathcal{F}(x_1,x_2,s)$ into \begin{eqnarray*} \mathcal{F}_{\alpha}(x_1, x_2, t)=t^2 A_{\alpha} + t B_{\alpha} +C_{\alpha} , \end{eqnarray*} with

Not being aware on that time of the fundamental work of [Komarov1981] and [Komarov and Kuznetsov1990], where the following deformations of the Kowalevski case were constructed and considered, Jurdjevic associated these systems to the Kirchhoff elastic problem, see ([Jurdjevic1999a]). The systems are defined by the Hamiltonians \begin{eqnarray*} H=\frac{1}{4}\left(M_1^2+M_2^2 + 2M_3^2\right) +\gamma_1 \end{eqnarray*} where the deformed Poisson structures $ \{\cdot, \cdot\}_{\tau}$ are defined by \begin{eqnarray*} \{M_i, M_j\}_{\tau}=\epsilon_{ijk}M_k,\quad \{M_i, \gamma_j\}_{\tau}=\epsilon_{ijk}\gamma_k,\quad \{\gamma_i, \gamma_j\}_{\tau}=\tau \epsilon_{ijk}M_k, \end{eqnarray*} and where the deformation parameter takes values $ \tau = 0, 1, -1$. These structures correspond to $ e(3)$, $ so(4)$, and $ so(3,1)$ respectively. The classical Kowalevski case corresponds to the case $ \tau=0$. These systems have been rediscovered by several authors in the meantime. Here, we are giving explicit formulae in theta-functions for the solutions of these systems.

Denote \begin{eqnarray*} e_1&=&x_1^2-(\gamma_1 + i\gamma_2) +\tau\\ e_2&=&x_2^2-(\gamma_1 - i\gamma_2) +\tau, \end{eqnarray*} where \begin{eqnarray*} x_{1,2}=\frac{M_1\pm i M_2}{2}. \end{eqnarray*} The integrals of motion \begin{eqnarray*} I_1&=&e_1 e_2\\ I_2&=&H\\ I_3&=&\gamma_1 M_1 + \gamma_2 M_2 + \gamma_3 M_3\\ I_4&=&\gamma_1^2 + \gamma_2^2+\gamma_3^2 + \tau (M_1^2+M_2^2+M_3^2) \end{eqnarray*} may be rewritten in the form \begin{eqnarray*} k^2&=&I_1=e_1 \cdot e_2\\ M_3^2&=&e_1 + e_2 + \hat E(x_1,x_2)\\ -M_3\gamma_3&=&-x_2e_1-x_1e_2 + \hat F(x_1, x_2)\\ \gamma_3^2&=&x_2^2e_1+x_1^2e_2 + \hat G(x_1,x_2), \end{eqnarray*} where

Let us point out that the previous consideration does not establish an isomorphism between the Kowalevski top and the Jurdjevic elastica. It does not provide a coordinate transformation which would map the former to the latter. Nevertheless, the previous Lemma opens a possibility to integrate the latter system along the same scheme used for the former system: the generalized K"otter trick is related to discriminatly separable polynomials, see [Dragović2010], and thus applicable to the Jurdjevic elasticae as well, see [Dragović and Kukić2014b].

More explicitly, we apply the generalized K"otter transformation derived in [Dragović2010] to obtain the expressions for $ M_i,\gamma_i$ in terms of $ P_i$ and $ P_{ij}$- functions for $ i,j=1,2,3$. A generalization of the K"otter transformation which provides commuting separated variables for the above systems was performed in [Komarov and Kuznetsov1990], [Komarov et al.2003]. First, we rewrite the equations of motion for Jurdjevic elasticae:

Now we introduce the following notation: \begin{eqnarray*} R(x_1,x_2) &=&\hat E x_1 x_2+ \hat F (x_1+x_2) +\hat G,\\ R_1(x_1,x_2) &=&\hat E \hat G - \hat F^2,\\ P(x_i) &=& \hat E x_i^2 +2 \hat F x_i +\hat G, \quad i=1,\,2. \end{eqnarray*}

Denote by $ m_i$ the zeros of polynomial $ f$ and \begin{eqnarray*} P_i=\sqrt{(s_1-m_i)(s_2-m_i)}\quad\,i=1,\,2,\,3, \end{eqnarray*}

\begin{eqnarray*} P_{ij}=P_iP_j\Big(\frac{\dot s_1}{(s_1-m_i)(s_1-m_j)}+\frac{\dot s_2}{(s_2-m_i)(s_2-m_j)}\Big) \end{eqnarray*} One can easily get

Finally, we obtain

The formulae expressing $ P_i, P_{ij}$ in terms of the theta-functions are given [Kowalevski1889]. This gives the explicit formulae for the elasticae.

4.1. Higher-Dimensional Generalizations of Rigid Body Dynamics.

In 1966, in his seminal paper [Arnold1966], Arnold observed that two very important examples of the equations of motion, the ones of the Euler top and the Euler equations of the motion of inviscid incompressible fluid can be seen in a unified way and interpreted as the equations of the geodesic flows on a corresponding Lie group. The Riemannian metric is given by the kinetic energy. In the case of the Euler top, the Lie group is $ SO(3)$ and the Riemannian metric, given by the Hamiltonian $ 2H=\langle M, \Omega\rangle$ is left invariant. In the case of the fluid flow, the Lie group is a group of the volumepreserving diffeomorphisms and the metric is right-invariant. Starting from that observation, Arnold derived the equations of the geodesic flows of a left invariant metric on an arbitrary Lie group--and the Euler-Arnold equations emerged. The left invariance of the metric implies, for example, that the equations of the Euler top are written in the Lax form $ \dot M=[M,\Omega]$, and hence one gets the family of the first integrals $ tr(M^k)$. The importance of Arnold's result is highlighted by the fact that many of the equations that appear in Physics can be represented as the Euler-Arnold equations.

The first ideas for constructing the higher-dimensional generalizations of the Euler top go back to the XIX century. Using some ideas of Cayley, Frahm presented the equations of the $ n$-dimensional Euler top in 1874. He also constructed the family of the first integrals. However, the number of the first integrals was not enough to prove the integrability for $ n> 4$ (see [Frahm1874], [Schottky1891]). In [Manakov1976] (not being aware of the results of Frahm) found an L-A pair for a wider class of metrics on $ SO(n)$ given by $ M_{ij}=\frac{a_i-a_j}{b_i-b_j}\Omega_{ij}$, and showed that this class belongs to the class considered by [Dubrovin1977]. Hence, the solutions can be expressed in theta functions.

Arnold's observation was a starting point for a wide class of generalizations of the rigid body motion. For some of them see for example ([Belokolos et al.1994], [Fedorov and Kozlov1995], [Trofimov and Fomenko1995]) and references therein.

Let us consider motion of $ N$ points in $ \mathbb{R}^n$ such that the distance between each two of them is constant in time. As an analogy with the three-dimensional case, we have two reference frames: the fixed and the moving ones. In the moving reference frame, the velocity of the point $ A$ is: \begin{eqnarray*} V_A(t)=B^{-1}\dot{q}_A(t)=B^{-1}\dot{B}Q_A=\Omega(t)Q_A \end{eqnarray*} where again $ Q_A$ represents the radius vector of the point $ A$, and $ \Omega$ is skew-symmetric matrix ($ \Omega\in so(n)$) representing the angular velocity of the body in the moving reference frame. The angular momentum is a skew-symmetric matrix defined by \begin{eqnarray*} M&=&\int_B (VQ^t-QV^t)dm=\int_B (\Omega QQ^t-QQ^t\Omega^t)dm\\ &=&\int_B (\Omega QQ^t+QQ^t\Omega)dm=\Omega I+I\Omega, \end{eqnarray*} where $ I=\int_B QQ^tdm$ is a constant symmetric matrix called the mass tensor of the body (see [Fedorov and Kozlov1995]) and integration goes over the body $ B$.

If one chooses the basis in which $ I=\diag(I_1,\ldots,I_n)$, the coordinates of angular momentum are $ M_{ij}=(I_i+I_j)\Omega_{ij}$.

The kinetic energy is \begin{eqnarray*} T=\frac12\int_B \langle \dot{Q},\dot{Q}\rangle dm=\frac12\int_B \langle \Omega Q,\Omega Q\rangle dm. \end{eqnarray*} Since it is a homogeneous quadratic form of angular velocity $ \Omega$, one has $ \langle\frac{\partial T}{\partial\Omega}, \Omega\rangle=2T$ where $ \langle A, B\rangle =-\frac12 Trace(AB)$ is an invariant scalar product on $ so(n)$. One gets \begin{eqnarray*} \frac{\partial T}{\Omega_{kl}}=\sum_{m}(\Omega_{km}I_{ml}+I_{km}\Omega_{ml}), \end{eqnarray*} or $ \frac{\partial T}{\partial Q}=M$ and finally \begin{eqnarray*} T=\frac{1}{2}\langle M, \Omega\rangle. \end{eqnarray*}

The Lie group $ E(3)$ can be regarded as a semidirect product of the Lie groups $ SO(3)$ and $ \mathbb{R}^3$. The product in the group given by \begin{eqnarray*} (A_1,r_1)\cdot(A_2,r_2)=(A_1A_2, r_1+A_1r_2) \end{eqnarray*} corresponds to the composition of two isometric transformations of the Euclidian space. The Lie algebra $ e(3)$ is a semidirect product of $ \mathbb{R}^3$ and $ so(3)$. Using isomorphism between the Lie algebras $ so(3)$ and $ \mathbb{R}^3$, given by (2.1) , one concludes that $ e(3)$ is also isomorphic to the semidirect product $ s=so(3)\times_{ad} so(3)$. The commutator in $ s$ is given by: \begin{eqnarray*} [(a_1,b_1),(a_2,b_2)]=([a_1,a_2],[a_1,b_2]+[b_1,a_2]). \end{eqnarray*}

One concludes, that there are two natural higher-dimensional generalizations of Eq. (2.2) . The first one is on the Lie algebra $ e(n)$ that is a semidirect product of $ so(n)$ and $ \mathbb{R}^n$. The other one is on semidirect product $ s=so(n)\times_{ad}so(n)$.

4.2. The Heavy Rigid Body Equations on $ e(n)$.

The Euler-Arnold equations of motion of a heavy rigid body fixed at a point on $ e(n)$ are (see [Belyaev1981], [Trofimov and Fomenko1995], [Jovanović2007] and references therein):

The $ n$-dimensional Lagrange top on $ e(n)$ is defined by Belyaev in [Belyaev1981] by conditions:

4.3. The Heavy Rigid Body Equations on $ s=so(n)\times_{ad}so(n)$.

The equations of the motion of a rigid body on semidirect product $ s=so(n)\times_{ad}so(n)$ were given by [Ratiu1982]:

Ratiu proved that Eq. (4.3) are Hamiltonian in the Lie-Poisson structure on coadjoint orbits of group $ S$ given by:

In [Ratiu1982], the Lagrange case was defined by $ I_1=I_2=a,\ I_3=\cdots=I_n=b,\ \chi_{12}=-\chi_{21}\ne 0,\ \chi_{ij}=0,\ (i,j)\notin \{(1,2), (2,1)\}$. The completely symmetric case was defined there by $ I_1=\cdots=I_n=a$, where $ \chi\in so(n)$ is an arbitrary constant matrix. It was shown in [Ratiu1982] that Eq. (4.3) in these cases could be represented by the following L-A pair: \begin{eqnarray*} \frac d{dt} (\lambda^2C+\lambda M+\Gamma)=[\lambda^2C+\lambda M+\Gamma, \lambda \chi+\Omega], \end{eqnarray*} where in the Lagrange case $ C=(a+b)\chi$, and in the symmetric case $ C=2a\chi$.

4.4. Four-Dimensional Rigid Body Motion.

To any $ 3\times 3$ skew-symmetric matrix one assigns one vector in three-dimensional space using isomorphism between $ \mathbb{R}^3$ and $ so(3)$. Using the the isomorphism between $ so(4)$ and $ so(3)\times so(3)$, one can assign two three-dimensional vectors $ A_1$ and $ A_2$ to $ (4\times4)$-skew-symmetric matrix $ A$.

Vectors $ A_1$ and $ A_2$ are defined by: \begin{eqnarray*} A_1=\frac{A_++A_-}{2},\quad A_2=\frac{A_+-A_-}{2}, \end{eqnarray*} where $ A_+,A_-\in \mathbb{R}^3$ correspond to $ A_{ij}\in so(4)$ according to:

By direct calculations, we check that vectors $ 2A_1\times B_1$ and $ 2A_2\times B_2$ correspond to commutator $ [A,B]$, if vectors $ A_1, A_2$ and $ B_1$, $ B_2$ correspond to $ A$ and $ B$ respectively.

Consequently, equations of motion (4.3) on $ so(4)\times so(4)$ can be written as:

Here we observe a complete analogy with the three-dimensional case. For example, the moment of inertia with respect to $ OZ$ axis $ I_{33}=\int_B (X^2+Y^2)dm$ consists of two addend $ \int_B X^2 dm$ and $ \int_B Y^2 dm$ that are diagonal elements of the mass tensor of the body.

For vectors $ M_+$ and $ M_-$ one has \begin{eqnarray*} M_+&=&\big((I_2+I_3)\Omega_{+}^1, (I_3+I_2)\Omega_+^2, (I_3+I_1)\Omega_+^3\big)=I_+\Omega_+\\ M_-&=&\big((I_1+I_4)\Omega_{-}^1, (I_2+I_4)\Omega_-^2, (I_3+I_4)\Omega_-^3\big)=I_-\Omega_-. \end{eqnarray*} Finally, one can calculate

At a glance it looks that (4.7) are equations of motion of two independent three-dimensional rigid bodies. However, the formulas (4.8) show that they are not independent and that each of $ M_1$, $ M_2$ depends on both $ \Omega_1$ and $ \Omega_2$.

4.5. The Lagrange Bitop: Definition and a Lax Representation.

Generalizing the Lax representation of the Hess-Appel'rot system, a new completely integrable four-dimensional rigid body system is established in [Dragovi'c and Gaji'c2001]. A detailed classical and algebro-geometric integration were presented in [Dragovi'c and Gaji'c2004].

The Lagrange bitop is a four-dimensional rigid body system on the semidirect product $ so(4)\times_{ad} so(4)$ defined by (see [Dragovi'c and Gaji'c2001], [Dragovi'c and Gaji'c2004]):

We have the following proposition:

Let us briefly analyze the spectral properties of the matrices $ L(\lambda)$. The spectral polynomial $ p(\lambda, \mu )=\det \left ( L(\lambda )-\mu \cdot 1\right)$ has the form \begin{equation*} p(\lambda , \mu )=\mu ^4+P(\lambda )\mu ^2 +[Q(\lambda )]^2, \end{equation*} where

System (4.3) , (4.9) doesn't fall in any of the families defined by [Ratiu1982] and together with them it makes complete list of systems with the $ L$ operator of the form \begin{eqnarray*} L(\lambda)=\lambda ^2C+\lambda M + \Gamma . \end{eqnarray*} More precisely, if $ \chi_{12}\ne 0$, then the Euler-Poisson Eq. (4.3) could be written in the form (4.10) (with arbitrary $ C$) if and only if Eq. (4.3) describe the generalized symmetric case, the generalized Lagrange case or the Lagrange bitop, including the case $ \chi_{12}=\pm \chi_{34}$ ([Dragovi'c and Gaji'c2001]).

4.6. Four-Dimensional Hess-Appel'rot Systems.

The starting point for construction of generalization of the Hess- Appel'rot system was Zhukovski's geometric interpretation given in Sect. 2.5. Having it in mind, in [Dragovi'c and Gaji'c2006] the higher-dimensional Hess-Appel'rot systems are defined. First we will consider the four-dimensional case on $ so(4)\times so(4)$. We will consider metric given with $ \Omega=JM+MJ$.

The invariant surfaces are determined in the following lemma.

Thus, in the four-dimensional Hess-Appel'rot case, there are two invariant relations

Let us now present another definition of the four-dimensional Hess- Appel'rot conditions, starting from a basis where the matrix $ J$ is diagonal in.

Let $ \tilde J=\diag(\tilde J_1, \tilde J_2, \tilde J_3, \tilde J_4)$.

One can calculate the spectral polynomial for the four-dimensional Hess-Appel'rot system: \begin{eqnarray*} p(\lambda, \mu)=\det(L(\lambda)-\mu\cdot 1)= \mu^4+P(\lambda)\mu^2+Q(\lambda)^2, \end{eqnarray*} where \begin{eqnarray*} P(\lambda)&=&a\lambda^4+b\lambda^3+c\lambda^2+d\lambda+e\\ Q(\lambda)&=&f\lambda^4+g\lambda^3+h\lambda^2+i\lambda+j \end{eqnarray*}

\begin{eqnarray*} a&=&C_{12}^2+C_{34}^2,\\ b&=&2C_{12}M_{12}+2C_{34}M_{34}(=0),\\ c&=&M_{13}^2+M_{14}^2+M_{23}^2+M_{24}^2+M_{12}^2+M_{34}^2+2C_{12}\Gamma _{12}+2C_{34}\Gamma_{34},\\ d&=&2\Gamma _{12}M_{12}+2\Gamma _{13}M_{13}+2\Gamma _{14}M_{14}+2\Gamma _{23} M_{23}+2\Gamma _{24}M_{24}+2\Gamma _{34}M_{34}\\ e&=&\Gamma _{12}^2+\Gamma _{13}^2+\Gamma _{14}^2+\Gamma _{23}^2+\Gamma _{24}^2 +\Gamma _{34}^2,\\ f&=&C_{12}C_{34}\\ g&=&C_{12}M_{34}+C_{34}M_{12}(=0),\\ h&=&\Gamma _{34}C_{12}+\Gamma_{12}C_{34}+M_{12}M_{34}+M_{23}M_{14}-M_{13}M_{24},\\ i&=&M_{34}\Gamma _{12}+M_{12}\Gamma _{34}+M_{14}\Gamma _{23}+M_{23}\Gamma _{14}- \Gamma _{13}M_{24}-\Gamma _{24}M_{13},\\ j&=&\Gamma _{34}\Gamma _{12}+\Gamma _{23}\Gamma _{14}-\Gamma _{13}\Gamma _{24}. \end{eqnarray*} In the standard Poisson structure on the semidirect product $ so(4)\times so(4)$ the functions $ d,e,i,j$ are Casimir functions, $ c, h$ are first integrals, and $ b=0, g=0$ are the invariant relations. As we already mentioned general orbits of co-adjoint action are eight-dimensional, thus for complete integrability one needs four independent integrals in involution.

4.7. The $ n$-Dimensional Hess-Appel'rot Systems.

In [Dragovi'c and Gaji'c2006] we introduced also Hess-Appel'rot systems of arbitrary dimension.

By diagonalizing the matrix $ J$, we come to another definition

As in the dimension four, there is an equivalence of the definitions.

The following theorem gives a Lax pair for the $ n$-dimensional Hess-Appel'rot system.

4.8. Classical Integration of the Four-Dimensional Hess-Appel'rot System.

Detailed classical and algebro-geometric integration procedures for the four-dimensional Hess-Appel'rot case are presented in [Dragovi'c and Gaji'c2006]. Here again Eq. (4.7) are useful for classical integration. We have: \begin{eqnarray*} \chi_1=\left(0,0,-\frac12 (\chi_{12}+\chi_{34})\right),\quad \chi_2=\left(0,0,-\frac12 (\chi_{12}-\chi_{34})\right). \end{eqnarray*} Integrals of the motion are

Let us introduce coordinates $ K_i$ and $ l_i$ as follows: \begin{eqnarray*} M_{(i)1}=K_i\sin l_i,\quad M_{(i)2}=K_i\cos l_i,\qquad i=1,2. \end{eqnarray*} From Eqs. (4.17) , (4.18) , using integrals (4.16) , we have \begin{eqnarray*} \dot\Gamma_{(1)3}^2=4(J_1+J_3)^2\left[(1-\Gamma_{(1)3}^2) (h_1-\frac2{J_1+J_3}\chi_{(1)3}\Gamma_{(1)3})-c_1^2\right]= P_3(\Gamma_{(1)3}). \end{eqnarray*} Thus $ \Gamma_{(1)3}$ can be solved by an elliptic quadrature. Also from the energy integral we have that \begin{eqnarray*} K_1^2=h_1-\frac2{J_1+J_3}\chi_{(1)3}\Gamma_{(1)3}. \end{eqnarray*} Since $ \tan l_1=\frac{M_{(1)1}}{M_{(1)2}}$, we have: \begin{eqnarray*} \dot l_1=-2(J_{13}+J_{24})K_2\sin l_2+\frac{2\chi_{(1)3}c_1}{K_1^2}. \end{eqnarray*} and \begin{eqnarray*} K_1^2\Gamma_{(1)2}^2-2c_1M_{(1)2}\Gamma_{(1)2}+c_1^2-M_{(1)1}^2(1-\Gamma_{(1)3}^2)=0. \end{eqnarray*} Similarly, one gets: \begin{eqnarray*} \dot\Gamma_{(2)3}^2&=&4(J_1+J_3)^2\left[(1-\Gamma_{(2)3}^2) (h_2-\frac2{J_1+J_3}\chi_{(2)3}\Gamma_{(2)3})-c_2^2\right]= P_3(\Gamma_{(2)3}),\\ K_2^2&=&h_2-\frac2{J_1+J_3}\chi_{(2)3}\Gamma_{(2)3},\\ \dot l_2&=&-2(J_{13}-J_{24})K_1\sin l_1+\frac{2\chi_{(2)3}c_2}{K_2^2}, \end{eqnarray*}

\begin{eqnarray*} K_2^2\Gamma_{(2)2}^2-2c_2M_{(2)2}\Gamma_{(2)2}+c_2^2-M_{(2)1}^2(1-\Gamma_{(2)3}^2)=0. \end{eqnarray*}

From the previous considerations one concludes that integration of the four-dimensional Hess-Appel'rot system leads to a system of two differential equations (for $ l_1$ and $ l_2$) of the first order and two elliptic integrals, associated with elliptic curves $ E_1$ and $ E_2$ defined by \begin{eqnarray*} E_i: y^2=P_i(x)=8A_ix^3-4B_ix^2-8A_ix-4C_i,\quad i=1,2 \end{eqnarray*} where \begin{eqnarray*} A_i=(J_1+J_3)\chi_{(i)3}, \quad B_i=(J_1+J_3)^2h_i,\quad C_i=(J_1+J_3)^2(c_i^2-h_i). \end{eqnarray*} This is a typical situation for the Hess-Appel'rot systems that additional integrations are required.

In [Dragovi'c and Gaji'c2006] the algebro-geometric integration procedure is presented. It is closely related to the integration of the Lagrange bitop.

5.1. The Classical Grioli Case.

By the above definition, a regular precession of three-dimensional rigid body motion assumes the existence of two vectors: a vector $ u$ fixed in the space and a vector $ U$ fixed in the body, such that the angle between them remains constant during the motion. Suppose the moving frame $ OE_1E_2E_3$ and the fixed frame $ Oe_1e_2e_3$ are chosen in a way that $ U=E_3$ and $ u=e_3$. Then the nutation angle $ \theta$ is constant. Grioli proved in [Grioli1947] that a regular precession is possible if $ \theta=\pi/2$ and $ \dot\psi=\dot\varphi=c=const.$ Expressed through the Euler angles $ \psi, \varphi, \theta$, the angular velocity $ \Omega$ and the vector $ u$ in the moving coordinate system are: \begin{eqnarray*} \Omega&=&(c\sin(ct), c\cos(ct), c),\\ u&=&(\sin (ct), \cos (ct),0) \end{eqnarray*} The unit vertical vector $ \Gamma$ is given by \begin{eqnarray*} &\displaystyle \Gamma=\big(\cos\sigma\sin(ct)+\sin\sigma\cos^2(ct),\\ &\displaystyle \cos\sigma\cos(ct)-\sin\sigma\cos(ct)\sin(ct),\sin\sigma\sin(ct)\big) \end{eqnarray*} where $ \sigma$ is constant (see [Rubanovskii1985], [Grioli1947], [Borisov and Mamaev2001]).

Grioli obtained the conditions for a nonvertical regular precession by plugging the last expressions into the Euler-Poisson equations (2.2) :

If $ \Gamma=u$ then $ I_{13}=0$, presenting the case of the regular vertical precession of the Lagrange top. Here $ \Omega$ and $ \Gamma$ are:

Thus, we see the Grioli case as a certain perturbation of the Lagrange top.

On the other hand, the Grioli precession can be obtained starting from the particular solutions (5.2) of the Lagrange top. By plugging the solutions for $ \Omega$ from (5.2) into the Euler-Poisson Eq. (2.2) and using the conditions (5.1) , from the first two equations, one determines $ \Gamma_1$ and $ \Gamma_2$. Then, one gets the expression for $ \Gamma_3$ from the differential equation for $ \Gamma_1$. Quite miraculously, the remaining differential equations are identically satisfied then.

In the next Section we will construct a particular solution of the fourdimensional Lagrange top first. Then, by using the pattern described above, we will construct a four-dimensional analogue of the Grioli precession.

5.2. Four-Dimensional Grioli Case.

For $ n=4$ the Lagrange top on $ e(4)$ defined by Belyaev (4.2) is given by: $ I=\diag(I_1, I_1, I_1, I_4)$ and $ X=(0,0,0,x_4)$. Thus, the Eq. (4.1) become:

To construct a particular solution of Eq. (5.3) , let us start with the case when $ x_4=0$. This is the symmetric Euler case. Let us fix the values of the first three linear integrals $ \Omega_{12}=-a$, $ \Omega_{13}=-b$, $ \Omega_{23}=-c$. Then the second three equations of (5.3) are of the form: \begin{eqnarray*} \left(\begin{array}{l} \dot{x}\\ \dot{y}\\ \dot{z} \end{array} \right) = \left(\begin{array}{lll} 0&\quad -a&\quad -b\\ a&\quad 0&\quad -c\\ b&\quad c&\quad 0 \end{array} \right) \left(\begin{array}{l} x\\ y\\ z \end{array} \right). \end{eqnarray*} The general solution of this system is given by:

It is important to observe that the angular velocity is decomposed into two components: $ \Omega=\Omega_1+\Omega_2$. The first component $ \Omega_1=\Omega_{12}E_1\wedge E_2+\Omega_{13}E_1\wedge E_3+\Omega_{23}E_2\wedge E_3$ represents a rotation in the three-dimensional space generated by the unit vectors $ E_1$, $ E_2$ and $ E_3$ and it is constant in the body.

The remaining component, $ \Omega_2=\Omega_{14}E_1\wedge E_4+ \Omega_{24}E_2\wedge E_4+\Omega_{34}E_3\wedge E_4$ can be written more explicitly as: \begin{eqnarray*} {\Omega_{2}=\left(\begin{matrix}0&\quad 0&\quad 0&\quad ac\cos(\lambda t)-b\lambda\sin(\lambda t)\\ 0&\quad 0&\quad 0&\quad -ab\cos(\lambda t)-c\lambda\sin(\lambda t)\\ 0&\quad 0&\quad 0&\quad -(b^2+c^2)\cos(\lambda t)\\ {\begin{matrix}-ac\cos(\lambda t)\\ +b\lambda\sin(\lambda t)\end{matrix}}&\quad {\begin{matrix}ab\cos(\lambda t)\\ +c\lambda\sin(\lambda t)\end{matrix}}&\quad (b^2+c^2)\cos(\lambda t)&\quad 0 \end{matrix}\right)} \end{eqnarray*} Thus, $ \Omega_{2}$ satisfies the Poisson equations \begin{eqnarray*} \dot{\Omega}_2=[\Omega_2,\Omega]. \end{eqnarray*} Consequently, $ \Omega_2$ represents a rotation of three-dimensional subspace fixed in the space. One can check that $ \Gamma$ is orthogonal to it. Note also that the vector $ \Gamma$ is orthogonal to the vector $ E_4$, so, as in three-dimensional case, we have here a case of the vertical precession.

In the previous subsection, we explained the procedure how to construct the classical Grioli precession in three-dimensional case, starting from the vertical precession of the Lagrange top. Let us apply here the same pattern to get a four-dimensional analogue of the Grioli case.

Denote by $ u$ a non-vertical vector fixed in the space. We are looking for a motion such that there exists a vector $ U$ fixed in the body, with the property that the angle between $ u$ and $ U$ is constant in time. Let us, by analogy with the three-dimensional case, fix a moving frame such with $ E_4=U$ and a fixed frame having $ e_4=u$. Then, the matrix $ I$ is not necessary diagonal any more.

More explicitly, the equations of motion (4.1) are:

Our next goal is to construct a particular solution of the four-dimensional Grioli case. We start from the particular solution (5.5) of the four-dimensional Lagrange top. Our specific choice of the moving frame with the axis $ E_4$ coinciding with the direction of the vector $ U$, motivates us to assume the particular solution for the $ \Omega$-s for the four-dimensional Grioli case to be the same as for the above particular solution of the four-dimensional Lagrange top. By plugging these expressions for $ \Omega$-s into the differential equations for $ \Omega$'s from (5.6) , one gets the expressions for the $ \Gamma_1$, $ \Gamma_2$, and $ \Gamma_3$. Then, from one of the three differential equations, say the one for $ \Gamma_1$, one gets the expression for $ \Gamma_4$. Finally, one needs to check if the remaining differential equations are identically satisfied. These compatibility conditions are satisfied again miraculously, under a very mild constraint on the parameters: if and only if $ b^2+c^2=1$, or equivalently if and only if $ \lambda=\sqrt{1+a^2}$. Summarizing, we get the following theorem:

6.1. Integrable Cases.

We will list the known completely integrable cases (for details see for example [Borisov and Mamaev2001]).

The first nontrivial integrable case of equations (6.1) was discovered by Kirchhoff in 1870 (see [Kirchhoff1874]). It is defined by the conditions:

• Kirchhoff's case (1870): \begin{eqnarray*} A=\diag(a_1, a_1, a_3),\quad B=\diag(b_1, b_1, b_3),\quad C=\diag(c_1, c_1, c_3). \end{eqnarray*} An additional integral is $ F_4=M_3$. It is analogous to the Lagrange case of motion of a heavy rigid body fixed at a point.
• The first Clebsch case (1871): \begin{eqnarray*} A=\diag (a,a,a),\quad B=0, \end{eqnarray*} The additional integral is: \begin{eqnarray*} F_4=a\langle C\vvec M, \vvec M\rangle-\det(C)\langle C^{-1}\vvec\Gamma, \vvec\Gamma\rangle. \end{eqnarray*}
• The second Clebsch case (1871):

  \begin{eqnarray} &&A=\diag(a_1, a_2, a_3),\quad B=0,\quad C=\diag(c_1,c_2,c_3)\nonumber\\ &&\qquad\frac{c_2-c_3}{a_1}+\frac{c_3-c_1}{a_2}+\frac{c_1-c_2}{a_3}=0. \label{cleb} \end{eqnarray}

  (6.2)

  The conditions (6.2) are equivalent to: \begin{eqnarray*} \frac{c_2-c_3}{a_1(a_2-a_3)}=\frac{c_3-c_1}{a_2(a_3-a_1)}=\frac{c_1-c_2}{a_3(a_1-a_2)}=\theta, \end{eqnarray*} where $ a_1, a_2, a_3$ are pairwise distinct. The additional integral is: \begin{eqnarray*} F_4=\theta\langle \vvec M,\vvec M\rangle-\langle A\vvec\Gamma,\vvec\Gamma\rangle. \end{eqnarray*}

• The Steklov case (1893): \begin{eqnarray*} A&=&\diag(a_1, a_2, a_3),\quad B=\diag(\mu a_2a_3,\mu a_3a_1, \mu a_1a_2),\\ C&=&\diag(\mu^2a_1(a_2-a_3)^2,\mu^2a_2(a_3-a_1)^2,\mu^2a_3(a_1-a_2)^2) \end{eqnarray*} where $ \mu$ is a constant. The additional integral is: \begin{eqnarray*} F_4&=&\sum\limits_j (M_j^2-2\mu a_jM_j\Gamma_j)+\mu^2\big((a_2-a_3)^2\Gamma_1^2\\ &&+(a_3-a_1)^2\Gamma_2^2+(a_1-a_2)^2\Gamma_3^2\big). \end{eqnarray*}
• The Lyapunov case (1893): \begin{eqnarray*} A&=&\diag(1,1,1),\quad B=\diag(-2\mu d_1,-2\mu d_2, -2\mu d_3),\\ C&=&\diag(\mu^2(d_2-d_3)^2,\mu^2(d_3-d_1)^2,\mu^2(d_1-d_2)^2) \end{eqnarray*} The additional integral is: \begin{eqnarray*} F_4&=&\sum\limits_j d_jM_j^2+2\mu(d_2d_3M_1\Gamma_1+d_3d_1M_2\Gamma_2+d_1d_2M_3\Gamma_3)\\ &&+\, \mu^2(d_1(d_2-d_3)^2\Gamma_1^2+d_2(d_3-d_1)^2\Gamma_2^2+d_3(d_1-d_2)^2\Gamma_3^2) \end{eqnarray*}
• The Sokolov case ([Sokolov2001]): \begin{eqnarray*} a_1&=&a_2=1,\quad a_3=2,\quad b_{13}=\alpha,\quad b_{23}=\beta,\\ c_{12}&=&-4\alpha\beta,\quad c_{11}=4\beta^2,\quad c_{22}=4\alpha^2,\quad c_{33}=-4(\alpha^2+\beta^2), \end{eqnarray*} and additional integral is \begin{eqnarray*} F_4=(M_3-\alpha \Gamma_1-\beta \Gamma_2)^2P+Q^2, \end{eqnarray*} where \begin{eqnarray*} P&=&(\alpha^2+\beta^2)(M_3+2\alpha \Gamma_1+2\beta \Gamma_2)^2+(\beta M_1-\alpha M_2)^2\\ Q&=&\big[\alpha M_1+\beta M_2+(\alpha^2+\beta^2)\Gamma_3\big](M_3+2\alpha \Gamma_1+2\beta \Gamma_2)\\ &&+\,3(\beta M_1-\alpha M_2)(\beta \Gamma_1-\alpha \Gamma_2). \end{eqnarray*}

  Among completely integrable cases, let us mention two more cases: the Chaplygin first case that admits the additional integral when $ \langle \vvec M, \vvec\Gamma\rangle=0$, and the Chaplygin second case that, instead of an additional first integral, possesses an invariant relation and it is analogous to the Hess-Appel'rot case.

• The Chaplygin first case (1902): \begin{eqnarray*} A=\diag(a, a, 2a), \quad B=0,\quad C=\diag(c, -c, 0), \end{eqnarray*} On the symplectic leaf given by $ \langle \vvec M, \vvec\Gamma\rangle=0$, the equations admit an additional first integral: \begin{eqnarray*} F_4=\left(M_1^2-M_2^2+c\Gamma_3^2\right)^2+4M_1^2M_2^2. \end{eqnarray*}
• The Chaplygin second case (1897).

  Chaplygin's second case has an invariant relation, instead of a fourth first integral. It was defined by Chaplygin in 1897 (see [Chaplygin1976]). This system was also considered by Kozlov and Onischenko in [Kozlov and Onischenko1982]. It is defined by:

  \begin{eqnarray} A&=&\diag(a_1, a_2, a_3)\nonumber\\ &&b_{13}\sqrt{a_2-a_1}\mp(b_2-b_1)\sqrt{a_3-a_2}=0,\, b_{12}=0\nonumber\\ &&b_{13}\sqrt{a_3-a_2}\pm(b_3-b_2)\sqrt{a_2-a_1}=0,\, b_{23}=0\nonumber\\ &&c_{13}\sqrt{a_2-a_1}\mp(c_2-c_1)\sqrt{a_3-a_2}=0,\, c_{12}=0\nonumber\\ &&c_{13}\sqrt{a_3-a_2}\pm(c_3-c_2)\sqrt{a_2-a_1}=0,\, c_{23}=0. \label{cc} \end{eqnarray}

  (6.3)

  An invariant relation is: $ F_4=M_1\sqrt{a_2-a_1}\mp M_3\sqrt{a_3-a_2}=0$. One can easily observe a lot of similarities between the Kirchhoff and the Euler-Poisson equations and a parallelism between corresponding integrable cases. However, the problem of integrability for the Kirchhoff equations appeared to be much complicated. Some partial results of the nonexistence of an additional integral were given by [Kozlov and Onischenko1982], [Borisov1996], [Barkin and Borisov1989], [Sadetov2000]. The problem of classification of completely integrable cases is still open for the Kirchhoff equations.

6.2. Three-Dimensional Chaplygin's Second Case.

Conditions (6.3) may be regarded as analogy of the Hess-Appel'rot conditions in the case of motion of a heavy rigid body fixed at a point. We have shown that Hess-Appel'rot case can be considered as a perturbation of the Lagrange top.

Similarly, the Chaplygin case is a perturbation of the Kirchhoff case. If one chooses the basis where $ a_1=a_2$, the Chaplygin conditions become (see for example [Dragovi'c and Gajić2012], [Borisov and Mamaev2001]): \begin{eqnarray*} a_1=a_2,\quad a_{13}\ne 0,\quad B=\diag (b_1, b_1, b_3),\quad C=\diag (c_1, c_1, c_3). \end{eqnarray*}

The Hamiltonian becomes: \begin{eqnarray*} 2H&=&a_1(M_1^2+M_2^2)+a_3M_3+{2a_{13}M_1M_3}\\ &&+\,2b_1(M_1p_1+M_2p_2)+2b_3M_3p_3+c_1(p_1^2+p_2^2)+c_3p_3^2\\ &=&H_K+2a_{13}M_1M_3. \end{eqnarray*} Here $ H_K$ is the Hamiltonian for the Kirchhoff case. In the new coordinates the invariant relation is $ M_3=0$.

6.3. Four-Dimensional Kirchhoff and Chaplygin Cases.

In [Dragovi'c and Gajić2012] the four-dimensional generalizations of the Kirchhoff and Chaplygin cases are constructed on $ e(4$).

Let us consider the Hamiltonian equations with the Hamiltonian function: \begin{eqnarray*} 2H=\sum A_{ijkl}M_{ij}M_{kl}+2\sum B_{ijk}M_{ij}\Gamma_k+\sum C_{kl}\Gamma_k\Gamma_l \end{eqnarray*} in the standard Lie-Poisson structure on $ e(4)$ given by: \begin{eqnarray*} \{M_{ij}, M_{kl}\}=\delta_{ik}M_{jl}+\delta_{jl}M_{ik}-\delta_{il}M_{jk}-\delta_{jk}M_{il} \end{eqnarray*}

\begin{eqnarray*} \{M_{ij},\Gamma_k\}=\delta_{ik}\Gamma_j-\delta_{jk}\Gamma_i \end{eqnarray*} A four-dimensional Kirchhoff case should have two linear first integrals: $ M_{12}$ and $ M_{34}$. It is interesting that under such an assumption, the "mixed" term in the Hamiltonian is missing.

On $ e(4)$ the standard Lie - Poisson structure has two Casimir functions: \begin{eqnarray*} F_1&=&\Gamma_1^2+\Gamma_2^2+\Gamma_3^2+\Gamma_4^2,\\ F_2&=&(M_{13}\Gamma_4-M_{14}\Gamma_3+M_{34}\Gamma_1)^2+(M_{23}\Gamma_1+M_{12}\Gamma_{3}-M_{13}\Gamma_2)^2\\ &&+\,(M_{24}\Gamma_1-M_{14}\Gamma_2+M_{12}\Gamma_4)^2+(M_{23}\Gamma_4+M_{34}\Gamma_2-M_{24}\Gamma_3)^2 \end{eqnarray*} consequently, the general symplectic leaves are 8-dimensional. For complete integrability one needs four first integrals in involution. In [Dragovi'c and Gajić2012] it is proved that except Hamiltonian, the four-dimensional Kirchhoff case has two linear first integrals $ F_3=M_{12}$, $ F_4=M_{34}$ and one additional quadratic first integral: \begin{eqnarray*} F_5&=&a_1(M_{12}M_{34}+M_{14}M_{23}-M_{13}M_{24})^2\\ &&-\,c_1((M_{13}\Gamma_4-M_{14}\Gamma_3+M_{34}\Gamma_1)^2+(M_{23}\Gamma_4+M_{34}\Gamma_2-M_{24}\Gamma_3)^2)\\ &&-\,c_3((M_{23}\Gamma_1+M_{12}\Gamma_{3}-M_{13}\Gamma_2)^2+(M_{24}\Gamma_1-M_{14}\Gamma_2+M_{12}\Gamma_4)^2) \end{eqnarray*} So, we have

The final step in our program is construction of the four-dimensional analogue of the Chaplygin case. One may naturally expect that $ M_{12}$ and $ M_{34}$ would appear as the invariant relations. Starting from this assumption, we propose:

One can easily check that in this case $ M_{12}$ and $ M_{34}$ are indeed the invariant relations. Thus, we have constructed a system which is a four-dimensional analogue of the Chaplygin case. At the same time, the obtained system plays the role for the case of the Kirchhoff equations on $ e(4)$ analogue to the role of the four-dimensional Hess-Appel'rot system for the Euler-Poisson equations on $ so(4)\times so(4)$.

The rigid body dynamics is one of the very basic classical problems with enormous range of applications. The connections with several other areas of mathematics and mechanics such as the theory of Lie groups and algebras, differential and algebraic geometry, fluid mechanics etc. reflect its importance. There are still many important unsolved problems. Among those concerning integrability, we can mention in particular the question of classification of integrable cases of the Kirchhoff equations. As it was presented above, there is an amazing parallelism between the known integrable cases of the Euler-Poisson equations and the Kirchhoff equations. Nevertheless, the theory related to the Kirchhoff equations is far from being finalized as it is the case for the Euler- Poisson equations.