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There exists a lattice map from the set of pure uniformizable Andersont-motives to the set of lattices. It is not known what is the image and thefibers of this map. We prove a local result that sheds the first light to thisproblem 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 theCarlitz 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-motivesthemselves, and (b) from the set of Siegel matrices to the set of lattices. Theresult of the present paper gives that the size of a neighborhood, where wehave an isomorphism, depends on an element of the monodromy group. We do notknow whether there exists a universal neighborhood. Method of the proof:explicit solution of an equation describing an isomorphism between twot-motives by a method of successive approximations using a version of theHensel lemma.

Lattice map for Anderson T-motives: first approach