  The Toda lattice defined by the Hamiltonian $H={1\over 2} \sum_{i=1}^n p_i^2
+ \sum_{i=1}^{n-1} \nu_i e^{q_i-q_{i+1}}$ with $\nu_i\in \{ \pm 1\}$, which
exhibits singular (blowing up) solutions if some of the $\nu_i=-1$, can be
viewed as the reduced system following from a symmetry reduction of a subsystem
of the free particle moving on the group $G=SL(n,\Real )$. The subsystem is
$T^*G_e$, where $G_e=N_+ A N_-$ consists of the determinant one matrices with
positive principal minors, and the reduction is based on the maximal nilpotent
group $N_+ \times N_-$. Using the Bruhat decomposition we show that the full
reduced system obtained from $T^*G$, which is perfectly regular, contains
$2^{n-1}$ Toda lattices. More precisely, if $n$ is odd the reduced system
contains all the possible Toda lattices having different signs for the $\nu_i$.
If $n$ is even, there exist two non-isomorphic reduced systems with different
constituent Toda lattices. The Toda lattices occupy non-intersecting open
submanifolds in the reduced phase space, wherein they are regularized by being
glued together. We find a model of the reduced phase space as a hypersurface in
${\Real}^{2n-1}$. If $\nu_i=1$ for all $i$, we prove for $n=2,3,4$ that the
Toda phase space associated with $T^*G_e$ is a connected component of this
hypersurface. The generalization of the construction for the other simple Lie
groups is also presented.
