The eigenvectors of the Hamiltonian $\mathscr{H}_N$ of $N$-sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero–Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.

We give a review about the integrability of complex analytical dynamical systems started with the works of Kovalevskaya, Liapounov and Painleve as well as by Picard and Vessiot at the end of the XIX century. In particular, we state a new result which generalize a theorem of Ramis and the author. This last theorem is itself a generalization of Ziglin's non-integrability theorem about the monodromy group of the first order variational equation. Also we try to point out some ideas about the connection of the above results with the Painleve property.

We investigate the possible integrable nonautonomous forms of a given class of mappings involving more than one dependent variable. These integrable discrete systems define "asymmetric" Painlevé equations. Our main tool of investigation is the application of the singularity confinement discrete integrability criterion. A new way of implementing it, first proposed for the singularity analysis of continuous systems, is also introduced.

In the framework of the thermodynamic formalism for dynamical systems [26] Selberg's zeta function [29] for the modular group $PSL(2,\mathbb{Z})$ can be expressed through the Fredholm determinant of the generalized Ruelle transfer operator for the dynamical system defined by the geodesic flow on the modular surface corresponding to the group $PSL(2,\mathbb{Z})$ [19]. In the present paper we generalize this result to modular subgroups $\Gamma$ with finite index of $PSL(2,\mathbb{Z})$. The corresponding surfaces of constant negative curvature with finite hyperbolic volume are in general ramified covering surfaces of the modular surface for $PSL(2,\mathbb{Z})$. Selberg's zeta function for these modular subgroups can be expressed via the generalized transfer operators for $PSL(2,\mathbb{Z})$ belonging to the representation of $PSL(2,\mathbb{Z})$ induced by the trivial representation of the subgroup $\Gamma$. The decomposition of this induced representation into its irreducible components leads to a decomposition of the transfer operator for these modular groups in analogy to a well known factorization formula of Venkov and Zograf for Selberg's zeta function for modular subgroups [34].

We suggest a new method of analysis of degeneracies in families of periodic solutions to an ODE, which is based upon the application of variational equations of higher order. The equation of oscillations of a satellite in the plane of its elliptic orbit (the Beletsky equation) is considered as a model problem. We study the degeneracies of arbitrary co-dimension in the families of its $2\pi$-periodic solutions and obtain the explicit formulas for them, which allows to localize the degeneracies with high accuracy and to give them a geometric interpretation.

We consider the double mathematical pendulum in the limit when the ratio of pendulums masses is close to zero and if the value of one of other system parameters is close to degenerate value (i.e. zero or infinity). We investigate homoclinic intersections, using Melnikov's method, and obtain an asymptotic formula for the homoclinic invariant in this case.

We study non-linear oscillations of a nearly integrable Hamiltonian system with one and half degrees of freedom in a neighborhood of an equilibrium. We analyse the resonance case of order one. We perform careful analysis of a small finite neighborhood of the equilibrium. We show that in the case considered the equilibrium is not stable, however, this instability is soft, i.e. trajectories of the system starting near the equilibrium remain close to it for an infinite period of time. We discuss also the effect of separatrices splitting occurring in the system. We apply our theory to study the motion of a particle in a field of waves packet.