Perhaps this is stupid question.

Let $\mathcal{M}$ be a symplectic manifold in $\mathbb{R}^4$ of codimension 2 with the symplectic 2-form $dx_1 \wedge dy_1 + dx_2 \wedge dy_2$. Suppose that $\mathcal{M}$ intersects the $(x_2,y_2)$-plane perpendicular in the origin. Does there exist a symplectic transformation $(x_2,y_2) \mapsto (\hat{x}_2,\hat{y}_2) $ such that for an open neighbourhood of the origin in $\mathcal{M}$ the $ \hat{x}_2,\hat{y}_2$ are constant zero?

Or is this complete nonsense?