Classical and quantum theory of finite-dimensional integrable systems

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1. The dressing chains are known to be an important nonlinear differential equation that includes the Painleve equations. These equations have a Lax representation by a second order square matrix, and the associated spectral curve becomes hyperelliptic. This enabled us to apply the technique of separation of variables, and to obtain a Hamiltonian representation of the dressing chains under a periodic boundary condition.2. A non-autonomous version of the SU(2) Calogero-Gaudin system was taken up an example of isomonodromic deformations on a torus. We applied the technique of separation of variables to rewrite this system into a Hamiltonian form. As a byproduct, we could find a connection of this matrix system with a scalar isomonodromic system.3. The Inozemtsev system is a deformation of the Calogero-Moser system retaining the classical integrability. We considered its quantum theory, and discovered that the quantum system has partial solvability (quasi-exact-solvability) at a discrete series of special values of one of the coupling constants.4. The moduli space of a class of rational functions is known to carry the structure of an integrable system. We constructed a new integrable system by replacing the rational functions by trigonometric or elliptic functions, and pointed out that these integrable systems provide very simple models of separation of variables.