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In this work, following [Bit15] and [Bit16a], we consider analytic singularvector fields in $(\mathbb{C}^{3},0)$ with an isolated and doubly-resonantsingularity of saddle-node type at the origin. Such vector fields come fromirregular two-dimensional differential systems with two opposite non-zeroeigenvalues, and appear for instance when studying the irregular singularity atinfinity in Painlev{\'e} equations (P\_j), j=I...V , for generic values of theparameters. Under suitable assumptions, we provide an analytic classificationunder the action of fibered diffeomorphisms, based on the study of the Stokesdiffeomorphisms obtained by comparing consecutive sectorial normalizing maps{\`a} la Martinet-Ramis / Stolovitch. These normalizing maps over sectorialdomains are obtained in the main theorem of [Bit16a], which is analogous to theclassical one due to Hukuhara-Kimura-Matuda for saddle-nodes in$\mathbb{C}^{3}$. We also prove that these maps are in fact the Gevrey-1 sumsof the formal normalizing map, the existence of which has been proved in[Bit15].

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