Doubly-resonant saddle-nodes in $(\mathbb{C}^{3},0)$ and the fixed singularity at infinity in the Painlev\'e equations (part II): sectorial normalization

In this work, following [Bit15], we consider analytic singular vector fields in $(\mathbb{C}^{3},0)$ with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlev{\'e} equations (P\_j), j=I...V , for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic nor-malization over sectorial domains, analogous to the classical one due to Hukuhara-Kimura-Matuda [HKM61] for saddle-nodes in $(\mathbb{C}^{2},0)$. We also prove that the normalizing map is essentially unique and weakly Gevrey-1 summable.