  Let $G$ be a complex Lie group and $\Lambda G$ denote the group of maps from
the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable
map $F$ from a manifold $M$ into $\Lambda G$, is said to be of \emph{connection
order $(_a^b)$} if the Fourier expansion in the loop parameter $\lambda$ of the
${\mathbb S}^1$-family of Maurer-Cartan forms for $F$, namely $F_\lambda^{-1}
\dd F_\lambda$, is of the form $\sum_{i=a}^b \alpha_i \lambda^i$. Most
integrable systems in geometry are associated to such a map. Roughly speaking,
the DPW method used a Birkhoff type splitting to reduce a harmonic map into a
symmetric space, which can be represented by a certain order $(_{-1}^1)$ map,
into a pair of simpler maps of order $(_{-1}^{-1})$ and $(_1^1)$ respectively.
Conversely, one could construct such a harmonic map from any pair of
$(_{-1}^{-1})$ and $(_1^1)$ maps. This allowed a Weierstrass type description
of harmonic maps into symmetric spaces. We extend this method to show that, for
a large class of loop groups, a connection order $(_a^b)$ map, for $a<0<b$,
splits uniquely into a pair of $(_a^{-1})$ and $(_1^b)$ maps. As an
application, we show that constant non-zero curvature submanifolds with flat
normal bundle of a sphere or hyperbolic space split into pairs of flat
submanifolds, reducing the problem (at least locally) to the flat case. To
extend the DPW method sufficiently to handle this problem requires a more
general Iwasawa type splitting of the loop group, which we prove always holds
at least locally.

Differential Geometry;Dynamical Systems