Symmetries, Conservation Laws and Integrability

The integrability of finite-dimensional dynamical systems is often associated with the existence of symmetries and first integrals. In the first talk, I shall explore how these concepts can be extended to differential-difference equations. We shall then consider further generalisations to dynamical systems and differential-difference equations in which the dependent variables take values in free associative algebras. Although this framework accommodates equations with matrix-valued variables, its direct application to systems defined over algebras with relations - such as quantum algebra - presents notable difficulties. A method to resolve these issues will be presented in the second talk.