With the above notation, we define

• (i)

  $\mathscr{L}_{Z}$ the set of sequences $\ell=\{\ell_{k}\}_{k\in\mathbb{N}}$ such that $\ell_{k}$ is a fibrewise closed section of $\mathfrak{L}^{\ast}$;

• ii)

  $\mathscr{U}_{Z}$ the set of pairs $(V,u)$ where $V\subseteq T^{\ast}U/\penalty\Lambda$ is a neighborhood of $Z$ and $u:V\rightarrow\mathbb{R}^{n}$ is a proper, smooth, $S^{1}$-invariant Lagrangian submersion with components $(u_{1},\ldots,u_{n})$ such that $u|_{Z}=\pi$ and $u_{1}=b_{1}$.