Algebraic constructions of integrable dynamical systems-extensions of the Volterra system

CONTENTS Introduction Chapter I. Integrable discretizations of the Korteweg-de Vries equation ??1. Integrable dynamical systems with quadratic non-linearity ??2. Integrable reductions of dynamical systems (1.3) ??3. Inregrable dynamical systems with an arbitrary degree of non-linearity Chapter II. The integrable integro-differential equation ??1. The integro-differential equation as a continuous limit of the family of dynamical systems ??2. The basic properties of the integro-differential equation (1.3) Chapter III. Integrable differential equations in algebras of smooth functions and in continuous associative algebras ??1. First integrals of differential equations connected with automorphisms of associative algebras ??2. Algebraic constructions of certain integrable equations ??3. Differential and integro-differential equations in algebras of functions ??4. A theorem on two commuting automorphisms and its applications ??5. Applications to the Euler equations on the direct sums of Lie algebras and Chapter IV. Integrable dynamical systems connected with simple Lie algebras ??1. Algebraic analogues of the Volterra system ??2. Integrable Hamiltonian perturbations of the generalized Toda lattices ??3. Differential equations admitting Lax representations with several spectral parameters ??4. Dynamical systems admitting a Lax representation and generalizing the Toda lattice References