With the above notation,

• i)

  Let $\mathscr{L}_{\bar{Z}^{\#}_{H}}$ the set of sequences of fibrewise closed sections of $\mathfrak{L}^{\ast}$ which vanish outside $\bar{Z}^{\circ}_{L}$ for some positive $L$ such that $2L<|\log|b|+\partial_{b_{2}}H|$ for every $b\in\Gamma$.

• ii)

  Let $\mathscr{U}_{\bar{Z}^{\#}_{H}}$ be the set of pairs $(V,u)$ where, for some positive $L$ and $L^{\prime}$ satisfying $2L^{\prime}<2L<|\log|b|+\partial_{b_{2}}H|$, $V$ is a neighborhood of $Z^{\circ}_{L}$ and $u:V\rightarrow\mathbb{R}^{n}$ is a smooth, $S^{1}$-invariant Lagrangian submersion, with cylindrical fibres, with components $(u_{1},u_{2},u_{3})$ such that $u|_{Z^{\circ}_{L}}=\pi^{\circ}$, $u|_{E_{L,L^{\prime}}}=\pi^{\circ}$ and $u_{1}=b_{1}$.

• iii)

  Let $\mathscr{H}_{\Delta}$ be the set of germs $H_{\Delta}$ of smooth functions $H$ defined on neighborhoods of $\Delta$.