  It is well-known that the Toda Theories can be obtained by reduction from the
Wess-Zumino-Novikov-Witten (WZNW) model, but it is less known that this WZNW
$\rightarrow$ Toda reduction is \lq incomplete'. The reason for this
incompleteness being that the Gauss decomposition used to define the Toda
fields from the WZNW field is valid locally but not globally over the WZNW
group manifold, which implies that actually the reduced system is not just the
Toda theory but has much richer structures. In this note we furnish a framework
which allows us to study the reduced system globally, and thereby present some
preliminary results on the global aspects. For simplicity, we analyze primarily
0 $+$ 1 dimensional toy models for $G = SL(n, {\bf R})$, but we also discuss
the 1 $+$ 1 dimensional model for $G = SL(2, {\bf R})$ which corresponds to the
WZNW $\rightarrow$ Liouville reduction.
