The Kovalevskaya exponents are sets of exponents that can be associated with a given nonlinear vector field. They correspond to the Fuchs' indices of the linearized vector field around particular scale invariant solutions. They were used by S.Kovalevskaya to prove the single-valuedness of the classical cases of integrability of the rigid body motion. In this paper, a history of the discovery and multiple re-discoveries of the Kovalevskaya exponents is given together with the modern use of Kovalevskaya exponents in integrability theory and nonlinear dynamics.

Results on the finite nonperiodic $A_n$ Toda lattice are extended to the Bogoyavlensky–Toda systems of type $B_n$ and $C_n$. The investigated areas include master symmetries, recursion operators, higher Poisson brackets and invariants. A conjecture which relates the degrees of higher Poisson brackets and the exponents of the corresponding Lie group is verified for these systems.

We present a fairly new and comprehensive approach to the study of stationary flows of the Korteweg–de Vries hierarchy. They are obtained by means of a double restriction process from a dynamical system in an infinite number of variables. This process naturally provides us with a Lax representation of the flows, which is used to find their bi-Hamiltonian formulation. Then we prove the separability of these flows making use of their bi-Hamiltonian structure, and we show that the variables of separation are supplied by the Poisson pair.

We review our findings on discrete Painleve equations with emphasis on the two direct methods we have proposed for their derivation: deautonomisation through singularity confinement and geometrical approach based on affine Weyl groups. The question of integrability of discrete Painleve equations is also addressed in terms of the existence of a Lax pair or of a description in the frame of the Grand Scheme. A list of discrete Painleve equations, as complete as possible but still not exhaustive, is also presented.

We describe a Poisson structure on some open subset of the relative Jacobian of hyperelliptic curves and use eigenvectors of Lax matrices to derive action variables for Moser systems.

The purpose of this note is to show that the Jacobi problem of geodesics on ellipsoid [1], [2], [3] may be reduced to a more simple case, namely, to the Clebsch problem [4]. The last one is the problem with quadratic nonlinearity and may be solved directly by using of Weber's approach [5] in terms of multi-dimensional theta functions.

The Kirchhoff analogy for elastic rods establishes the equivalence between the solutions of the classical spinning top and the stationary solutions of the Kirchhoff model for thin elastic rods with circular cross-sections. In this paper the Kirchhoff analogy is further generalized to show that the classical Kovalevskaya solution for the rigid body problem is formally equivalent to the solution of the Kirchhoff model for thin elastic rod with anisotropic cross-sections (elastic strips). These Kovalevskaya rods are completely integrable and are part of a family of integrable travelling waves solutions for the rod (Kovalevskaya waves). The analysis of homoclinic twistless Kovalevskaya rod reveals the existence of a three parameter family of solutions corresponding to the Steklov and Bobylev integrable case of the rigid body problem. Furthermore, the existence of these integrable solutions is discussed in conjunction with recent results on the stability of strips.

Images of the Liouville tori and three-dimensional isoenergetic surfaces are constructed in movable space of angular velocities and all possible types of these invariant sets are classified. The characteristic properties of the angular momentum and the angular velocity of the Kovalevskaya top are indicated.

We consider compositions of the transformations of the time variable and canonical transformations of the other coordinates, which map a completely integrable system into another completely integrable system. Change of the time gives rise to transformations of the integrals of motion and the Lax pairs, transformations of the corresponding spectral curves and $R$-matrices.