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

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