Somewhere dense orbit of abelian subgroup of diffeomorphisms maps acting on Cⁿ

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In this paper, we give a characterization for any abelian subgroup G of a lie group of diffeomorphisms maps of Cⁿ, having a somewhere dense orbit G(x), x in Cⁿ: G(x) is somewhere dense in Cⁿ if and only if there are f₁,....,f2n+1 in exp⁻¹(G) such that f₂ₙ₊₁ in vect(f₁,...,f₂ₙ) and Z.f₁(x)+....+Z.f₂ₙ₊₁(x) is dense subgroup of Cⁿ, where vect(f₁,....,f₂ₙ) is the vector space over R generated by f₁,....,f₂ₙ.