  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.
