Let $\mathcal{F}$ be a singular one-dimensional holomorphic foliation on a complex manifold $M$ of complex dimension $2$ and $p \in M$ a singular point for the foliation $\mathcal{F}$.
Assume that the eigenvalues $\lambda_1,\lambda_2$ of the foliation $\mathcal{F}$ at the singular point $p$ are both non-zero and satisfy $\frac{\lambda_1}{\lambda_2} \in ]0,+\infty[$ (the singular point $p$ lies in the Poincaré domain). In addition, assume that the foliation $\mathcal{F}$ at the singular point $p$ is locally conjugate to a Poincaré-Dulac normal form.
Under the assumptions above which of the following statements is correct? $$\frac{\lambda_1}{\lambda_2} \in \{n, \frac{1}{n}; n \in \mathbb{N}, n\geqslant2\}$$ or $$\frac{\lambda_1}{\lambda_2} \in \{n, \frac{1}{n}; n \in \mathbb{N}, n\geqslant1\}$$