Yury Fedorov

The paper revises the explicit integration of the classical Steklov–Lyapunov systems via separation of variables, which had been first made by F. Kötter in 1900, but was not well understood until recently. We give a geometric interpretation of the separating variables and then, applying the Weierstrass hyperelliptic root functions, obtain explicit theta-function solution to the problem. We also analyze the structure of poles of the solution on the Jacobian on the corresponding hyperelliptic curve. This enables us to obtain a solution for an alternative set of phase variables of the systems that has a specific compact form.
In conclusion we discuss the problem of integration of the Rubanovsky gyroscopic generalizations of the above systems.

We revise the solution to the problem of Hamiltonization of the $n$-dimensional Veselova non-holonomic system studied previously in [1]. Namely, we give a short and direct proof of the hamiltonization theorem and also show the trajectorial equivalence of the problem with the geodesic flow on the ellipsoid.

We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel–Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems.
Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.

We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid $Q$ that are cut out by algebraic surfaces in $\mathbb{R}^3$. Such geodesics are either connected components of real parts of spatial elliptic curves or of rational curves. Our approach is based on elements of the Weierstrass–Poncaré reduction theory for hyperelliptic tangential covers of elliptic curves, the addition law for elliptic functions, and the Moser–Trubowitz isomorphism between geodesics on a quadric and finite-gap solutions of the KdV equation. For the case of 3-fold and 4-fold coverings, some explicit formulas for the cutting algebraic surfaces are provided and some properties of the corresponding geodesics are discussed.

In the modern approach to integrable Hamiltonian systems, their representation in the Lax form (the Lax pair or the $L$–$A$ pair) plays a key role. Such a representation also makes it possible to construct and solve multi-dimensional integrable generalizations of various problems of dynamics. The best known examples are the generalizations of Euler's and Clebsch's classical systems in the rigid body dynamics, whose Lax pairs were found by Manakov [10] and Perelomov [12]. These Lax pairs include an additional (spectral) parameter defined on the compactified complex plane or an elliptic curve (Riemann surface of genus one). Until now there were no examples of $L$–$A$ pairs representing physical systems with a spectral parameter running through an algebraic curve of genus more than one (the conditions for the existence of such Lax pairs were studied in [11]).
In the given paper we consider a new Lax pair for the multidimensional Manakov system on the Lie algebra $so(m)$ with a spectral parameter defined on a certain unramified covering of a hyperelliptic curve. An analogous $L$–$A$ pair for the Clebsch–Perelomov system on the Lie algebra $e(n)$ can be indicated.
In addition, the hyperelliptic Lax pair enables us to obtain the multidimensional generalizations of the classical integrable Steklov–Lyapunov systems in the problem of a rigid body motion in an ideal fluid. The latter is known to be a Hamiltonian system on the algebra $e(3)$. It turns out that these generalized systems are defined not on the algebra $e(n)$, as one might expect, but on a certain product $so(m)+so(m)$. A proof of the integrability of the systems is based on the method proposed in [1].

It has been discovered a countable number of dynamic systems with an equal countable set of the first integrals and invariant measure. The found systems are a generalization of so-called Manakov's systems on $SO(n)$ algebra and the integrable Chaplygin's problem about ball rolling.