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The full Kostant-Toda (f-KT) lattice is a natural generalization of theclassical tridiagonal Toda lattice. We study singular structure of solutions ofthe f-KT lattices defined on simple Lie algebras in two different ways: throughthe $\tau$-functions and through the Kowalevski-Painlev\'e analysis. The$\tau$-function formalism relies on and is equivalent to the representationtheory of the underlying Lie algebras, while the Kowalevski-Painlev\'e analysisis representation independent and we are able to characterize all the terms inthe Laurent series solutions of the f-KT lattices via the structure theory ofthe Lie algebras. Through the above analysis we compactify the initialcondition spaces of f-KT lattice by the corresponding flag varieties, that isfixing the spectral parameters which are invariant under the f-KT flows, webuild a one to one correspondence between solutions of the f-KT lattices andpoints in the corresponding flag varieties. As all the important characters weobtain in the Kowalevski-Painlev\'e analysis are integral valued, results inthis paper are valid in any field containing the rational field.

On the full Kostant-Toda lattice and the flag varieties. I. The singular solutions