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Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to gradeone regular semisimple elements from non-equivalent Heisenberg subalgebras of aloop algebra $\G\otimes{\bf C}[\lambda,\lambda^{-1}]$ are studied. The gradedHeisenberg subalgebras containing such elements are labelled by the regularconjugacy classes in the Weyl group ${\bf W}(\G)$ of the simple Lie algebra$\G$. A representative $w\in {\bf W}(\G)$ of a regular conjugacy class can belifted to an inner automorphism of $\G$ given by $\hat w=\exp\left(2i\pi {\rmad I_0}/m\right)$, where $I_0$ is the defining vector of an $sl_2$ subalgebraof $\G$.The grading is then defined by the operator $d_{m,I_0}=m\lambda {d\overd\lambda} + {\rm ad} I_0$ and any grade one regular element $\Lambda$ from theHeisenberg subalgebra associated to $[w]$ takes the form $\Lambda = (C_++\lambda C_-)$, where $[I_0, C_-]=-(m-1) C_-$ and $C_+$ is included in an$sl_2$ subalgebra containing $I_0$. The largest eigenvalue of ${\rm ad}I_0$ is$(m-1)$ except for some cases in $F_4$, $E_{6,7,8}$. We explain how these Liealgebraic results follow from known results and apply them to constructintegrable systems.If the largest ${\rm ad} I_0$ eigenvalue is $(m-1)$, thenusing any grade one regular element from the Heisenberg subalgebra associatedto $[w]$ we can construct a KdV system possessing the standard $\W$-algebradefined by $I_0$ as its second Poisson bracket algebra. For $\G$ a classicalLie algebra, we derive pseudo-differential Lax operators for thosenon-principal KdV systems that can be obtained as discrete reductions of KdVsystems related to $gl_n$. Non-abelian Toda systems are also considered.Comment: 44 pages, ENSLAPP-L-493/94, substantial revision, SWAT-95-77. (use  OLATEX (preferred) or LATEX

Regular conjugacy classes in the Weyl group and integrable hierarchies