Vey theorem in infinite dimensions and its application to KdV

We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^{+},u_1^{-};u_2^{+},u_2^{-};\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.})\}$ with integrals $I_1, I_2,…$whichcanbewrittenas$I_j={1/2}|F_j|^2$,where$F_j:H\to \R^2$,$F_j(0)=0$for$j=1,2,…$\mathrm{.}Weassumethatthemaps$F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,\mathrm{.}\mathrm{.}\mathrm{.}):H\to H$, such that dF(0)=id$,$(F-id)$isa$\kappa$-smoothingmap($\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 $F_j^{\mathrm{\prime }}$ such that $F_j-F_j^{\mathrm{\prime }}=O(\mathrm{\mid }u{\mathrm{\mid }}^2)$ and each $\frac{1}{2}\mathrm{\mid }{F}_j^{\mathrm{\prime }}{\mathrm{\mid }}^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $F^{\mathrm{\prime }}:H\to H$, the germ $(F^{\mathrm{\prime }}-id)$ is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector $(\frac{1}{2}\mathrm{\mid }{F}_j^{\mathrm{\prime }}{\mathrm{\mid }}^2,j\ge 1)$. 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’.