# Integrable systems

We study interrelation between geometry of algebraic curves and their Jacobians and classical integrable dynamical systems. Rigid body systems with their natural higher dimensional generalisations and billiard systems within quadrics are chosen as main objects of the study. Among rigid body systems, two important classes are recognized, isoholomorphic systems and systems of Hess-Appel’rot type. Their dynamical and geometrical properies are studied. Integration procedure is completed based on deep facts from geometry of the Prym varieties of double unramified coverings - Mumfords relation.

For billiard systems, the thirty-years old programme of Griffiths and Harris of understanding higher-dimensional analogues of Poncelet-type problems and synthetic approach to higher genera addition theorems has been settled and completed in our last work.

Integrable systems solvable in the framework of the Quantum Inverse Scattering Method (QISM) can be classified by the underlying dynamical symmetry algebras. Among the simplest examples are the Gaudin models based on loop algebras and classical r-matrices. More complex quantum integrable systems correspond to more sophisticated infinite dimensional algebras: Yangians, quantum affine algebras, elliptic quantum groups, etc. Recent developments of quantum elliptic algebras have revealed a number of deep connections among these three types of algebraic structures present in quantum integrable systems.

The Algebraic Bethe Ansatz is a powerful algebraic tool which yields the spectrum and corresponding eigenstates in the applications of the QISM to the models for which highest weight type representations are relevant, like for example Gaudin models, quantum spin systems, etc. The Yang-Baxter relations and the QISM technique provide a natural choice of Hamiltonians in involution and their eigenvectors are constructed by Algebraic Bethe Ansatz. Recently we applied the algebraic Bethe ansatz to some elliptic quantum groups and Yangians further deformed by Jordanian twist.