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This paper concerns the solutions of the full Kostant-Toda (f-KT) hierarchyin the Hessenberg form and their reductions to the $\ell$-banded Kostant-Toda($\ell$-KT) hierarchy. We also study the f-KT hierarchy and the corresponding$\ell$-KT hierarchy on simple Lie algebras of type $A, B$ and $G$ based on rootspace reductions with proper Chevalley systems. Explicit formulas of thepolynomial solutions for the $\tau$-functions are also given in terms of theSchur functions and Schur's $Q$-functions.

On the full Kostant-Toda hierarchy and its $\ell$-banded reductions for the Lie algebras of type $A, B$ and $G$