  We define iteration over a two dimensional manifold as analog of iteration
over a path defined by Chen. We give several applications. Some of them include
constructions of non-abelian modular symbol for $SL(3,\Z)$ and for $SL_{2/K}$,
where $K$ is a real quadratic field. We construct the latter as generating
series of iterated integrals over a two-dimensional manifold of Hilbert modular
forms. We give a motivic interpretation of this non-abelian modular symbol.
Other applications give motivic interpretations of certain iterated completed
Dedekind zeta functions.
