Lie algebraic structures of (1+1)-dimensional Lax integrable systems

An approach of constructing isospectral flows Kl, nonisospectral flows σk and their implicit representations of a general Lax integrable system is proposed. By introducing product function matrices, it is shown that the two sets of flows and of related symmetries both constitute infinite-dimensional Lie algebras with respect to the commutator ⟦⋅,⋅⟧ given in this paper. Algebraic properties for some well-known integrable systems such as the AKNS system, the generalized Harry Dym system, and the n-wave interaction system are obtained as particular examples.