  There exists a lattice map from the set of pure uniformizable Anderson
t-motives to the set of lattices. It is not known what is the image and the
fibers of this map. We prove a local result that sheds the first light to this
problem and suggests that maybe this map is close to 1 -- 1. Namely, let $M(0)$
be a t-motive of dimension $n$ and rank $r=2n$ \ --- \ the $n$-th power of the
Carlitz module of rank 2, and let $M$ be a t-motive which is in some sense
"close" to $M(0)$. We consider the lattice map $M \mapsto L(M)$, where $L(M)$
is a lattice in $C^n$. We show that the lattice map is an isomorphism in a
"neighborhood" of $M(0)$. Namely, we compare the action of monodromy groups:
(a) from the set of equations defining t-motives to the set of t-motives
themselves, and (b) from the set of Siegel matrices to the set of lattices. The
result of the present paper gives that the size of a neighborhood, where we
have an isomorphism, depends on an element of the monodromy group. We do not
know whether there exists a universal neighborhood. Method of the proof:
explicit solution of an equation describing an isomorphism between two
t-motives by a method of successive approximations using a version of the
Hensel lemma.
