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Consider a 4-dim symplectic vector field $X$ on the symplectic manifold $(M, \omega)$ in $\mathbb{R}^4$ with $\omega= \sum_{i=1}^2 dy_i \wedge dx_i$. Moreover, the linear terms of $X$ are given by $y_1 \partial / \partial x_1 - x_1 \partial / \partial y_1$ (so not dependent on $(x_2,y_2)$). Let $X|_{\mathcal{N}}$ be the symplectic vector field restricted to $(\mathcal{N}, \omega|_{\mathcal{N}})$ a 2-dim symplectic submanifold of $(M, \omega)$. Furthermore, suppose that $T_0\mathcal{N}$ lies in the symplectic complement of the tangent space of the plane $(x_2,y_2)$. Does there exist a symplectic transformation such that in an open neighbourhood of zero the $(\mathcal{N}, \omega|_{\mathcal{N}})$ is given by the symplectic manifold $(\hat{\mathcal{N}}, \varpi)$ in $\mathbb{R}^2$ with

\begin{equation} \varpi= d\hat{y}_1 \wedge d\hat{x}_1 \end{equation}

and such that the linear terms of $X|_{\hat{\mathcal{N}}}$ are given by
\begin{equation} \hat{y}_1 \partial / \partial \hat{x}_1 - \hat{x}_1 \partial / \partial \hat{y}_1 \end{equation}

The existence of a 2-form follows by Darboux's theorem but I don't see how to construct a symplectic transformation such that the linear terms become as above.

Any help is welcome.

Thanks in advance.

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