Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum\_{i,j}{a\_{ij}}z\_i\frac{\partial}{\partial z\_j}$ be the linear term of $v$ and suppose that the matrix $a\_{ij}$ is invertible and all its eigenvalues have modulus different from $1$. Is it true that for some holomorphic coordinates $w\_i$ in a neighbourhood of $0$ we have $v=\sum\_{i,j}{a\_{ij}}w\_i\frac{\partial}{\partial w\_j}$?


If yes, where could I find such a statement? If not, what would be a counterexample? I am happy to assume that the eigenvalues of $A$ all have modulus less than $1$.



Holomorphic vector fields with a non-degenerate isolated zero
A relevant result, but not a complete answer.
Vladimir Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, p. 181:



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> A zero of a vector field is in the Poincare domain if the origin is not in the convex hull of the eigenvalues of the linearization.
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> A resonance of a zero of a vector field is an equation $\lambda\_i=\sum\_{j\ne i} m\_j \lambda\_j$ among eigenvalues of the linearization of the vector field about the zero, where $\{m\_j\}$ are integers, $m\_j\ge 0$, and $\sum\_j m\_j\ge 2$.
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> Theorem of Poincare: If the eigenvalues of the linearization of a formal analytic vector field are nonresonant, then it can be reduced to a linear vector field by a formal analytic change of variables. If the eigenvalues of the linear part of a holomorphic vector field at a singular point belong to the Poincare domain and are nonresonant, then the field is biholomorphically equivalent to its linear part in the
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