Lie algebras associated to systems of Dyson-Schwinger equations

We consider systems of combinatorial Dyson-Schwinger equations in the Connes-Kreimer Hopf algebra HI of rooted trees decorated by a set I. Let H(S) be the subalgebra of HI generated by the homogeneous components of the unique solution of this system. If it is a Hopf subalgebra, we describe it as the dual of the enveloping algebra of a Lie algebra g(S) of one of the following types:1.g(S) is an associative algebra of paths associated to a certain oriented graph.2.Or g(S) is an iterated extension of the Faà di Bruno Lie algebra.3.Or g(S) is an iterated extension of an infinite-dimensional abelian Lie algebra. We also describe the character groups of H(S).