Kac-Moody-Virasoro Symmetries and Related Conservation Laws

In this report, some important facts on the symmetries and conservation laws of high dimensional integrable systems are discussed. It is summarized that almost all the known (2+1)-dimensional integrable models possess the Kac-Moody-Virasoro (KMV) symmetry algebras. One knows that infinitely many partial differential equations may possess a same KMV symmetry algebra. It is found that the KMV symmetry groups can be explicitly obtained by using some direct methods. For some quite general variable coefficient nonlinear systems, their sufficient and necessary condition for the existence of the KMV symmetry algebra is they can be changed to the related known constant coefficient models. Finally, it is found that every one symmetry may be related to infinitely many conservation laws and then infinitely many models may possess a same set of infinitely many conservation laws.