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We consider the holomorphic normalization problem for a holomorphic vectorfield in the neighborhood of the product of a fixed point and an invarianttorus. Supposing that the vector field is a perturbation of a linear partaround the fixed point and of a rotation on the invariant torus (theunperturbed vector field is called the quasi-linear part of the perturbed one),it was shown by J.Aurouet that the system is holomorphically linearizable ifthere are no exact resonances in the quasi-linear part and if the quasi-linearpart satisfies to Brjuno's arithmetical condition. In the presence of exactresonances, a conjecture by Brjuno states that the system will still beholomorphically conjugated to a normal form under the same arithmeticalcondition and a strong algebraic condition on the formal normal form. Thisarticle proves this conjecture.

Normal form of holomorphic vector fields with an invariant torus under Brjuno's A condition