SYMMETRY ALGEBRAS AND GROUP INVARIANT SOLUTIONS OF THE KAWAMOTO-TYPE EQUATIONS

Starting from 10 sets of symmetries of the Kawamoto equation, one can get 10 hierarchies of corresponding integrable models. These hierarchi es possess the same recursion operator but they do not possess common symmetry structure. Only two hierarchies related to the kernel of inve rse recursion operators have the same time-independent symmetry struct ure as that of the Kawamoto equation. However, the symmetry algebras o f the other eight hierarchies are only isomorphic to some subalgebras of that of the Kawamoto equation. Especially, three hierarchies of the m possess six sets of time-independent noncommute symmetries while the other five hierarchies possess only four sets of time-independent but commute symmetries. Only one set of time-dependent symmetries for all ten hierarchies is obtained. Using the point symmetry algebra of the Kawamoto equation, various new complex group-invariant solutions (simi larity reductions) of the Kawamoto equation are also given.