  This paper deals with the category of nonlinear evolution equations (NLEEs)
associated with the spectral problem and provides an approach for constructing
their algebraic structure and $r$-matrix. First we introduce the category of
NLEEs, which composes of various positive order and negative order hierarchies
of NLEEs both integrable and non-integrable. The whole category of NLEEs
possesses a generalized Lax representation. Next, we present two different Lie
algebraic structures of the Lax operator, one of them is universal in the
category,i.e. independent of the hierarchy, while the other one is nonuniversal
in the hierarchy, i.e. dependent on the underlying hierarchy. Moreover, we find
that two kinds of adjoint maps are $r$-matrices under the algebraic structures.
In particular, the Virasoro algebraic structures without central extension of
isospectral and non-isospectral Lax operators can be viewed as reductions of
our algebraic structure. Finally, we give several concrete examples to
illustrate our methods. Particularly, the Burgers category is linearized when
the generator, which generates the category, is chosen to be independent of the
potential function. Furthermore, an isospectral negative order hierarchy in the
Burger's category is solved with its general solution. Additionally, in the KdV
category we find an interesting fact: the Harry-Dym hierarchy is contained in
this category as well as the well-known Harry-Dym equation is included in a
positive order KdV hierarchy.
