and 
, where 
is an effective field and 
and 
are given a suitable
asymptotic ordering 
.
Consider the mapping 
, where
. For 
, it is natural to ask how to solve the system 
. In this paper, we will
effectively characterize 
and
show how to compute a so called distinguished right inverse 
of 
.
We will also characterize the solution space of the homogeneous equation
.