Vey theorem in infinite dimensions and its application to KdV

We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$,$(F-id)$is a$\kappa$-smoothing map ($\kappa\geq 0$) and some other mild restrictions on$F$hold. Under these assumptions we show that the maps$F_j$may be modified to maps$F_j^\prime$such that$F_j-F_j^\prime=O(|u|^2)$and each$\frac12|F'_j|^2$still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism$F^\prime: H\to H$, the germ$(F^\prime-id)$is$\kappa$-smoothing, and each$I_j$is an analytic function of the vector$(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with$\kappa=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'.