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This paper deals with the category of nonlinear evolution equations (NLEEs)associated with the spectral problem and provides an approach for constructingtheir algebraic structure and $r$-matrix. First we introduce the category ofNLEEs, which composes of various positive order and negative order hierarchiesof NLEEs both integrable and non-integrable. The whole category of NLEEspossesses a generalized Lax representation. Next, we present two different Liealgebraic structures of the Lax operator, one of them is universal in thecategory,i.e. independent of the hierarchy, while the other one is nonuniversalin the hierarchy, i.e. dependent on the underlying hierarchy. Moreover, we findthat two kinds of adjoint maps are $r$-matrices under the algebraic structures.In particular, the Virasoro algebraic structures without central extension ofisospectral and non-isospectral Lax operators can be viewed as reductions ofour algebraic structure. Finally, we give several concrete examples toillustrate our methods. Particularly, the Burgers category is linearized whenthe generator, which generates the category, is chosen to be independent of thepotential function. Furthermore, an isospectral negative order hierarchy in theBurger's category is solved with its general solution. Additionally, in the KdVcategory we find an interesting fact: the Harry-Dym hierarchy is contained inthis category as well as the well-known Harry-Dym equation is included in apositive order KdV hierarchy.Comment: 24 pages, 0 figure

Category of nonlinear evolution equations, algebraic structure, and r-matrix