Vey theorem in infinite dimensions and its application to KdV

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We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space H={u=(u₁+,u₁-; u₂+,u₂-;....)} with integrals I₁, I₂,... which can be written as Iⱼ=1/2|Fⱼ|², where Fⱼ:H→ ², Fⱼ(0)=0 for j=1,2,.... We assume that the maps Fⱼ define a germ of an analytic diffeomorphism F=(F₁,F₂,...):H→ H, such that dF(0)=id, (F-id) is a κ-smoothing map (κ≥ 0) and some other mild restrictions on F hold. Under these assumptions we show that the maps Fⱼ may be modified to maps Fⱼ′ such that Fⱼ-Fⱼ′=O(|u|²) and each 12|F'ⱼ|² still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism F′: H→ H, the germ (F′-id) is κ-smoothing, and each Iⱼ is an analytic function of the vector (12|F'ⱼ|²,j≥1). Next we show that the theorem with κ=1 applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form `identity plus a 1-smoothing analytic map'.