There exists a proper submanifold $Q^{n}\subset(\mathbb{C}^{*})^{n+1}$ such that

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  $Q^{n}$ is embedded to $(\mathbb{C}^{*})^{n+1}$ symplectically, i.e. so that the restriction of the form (2) to $Q^{n}$ is a symplectic form.

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  $Q^{n}$ is isotopic to $H^{\circ}$ in $(\mathbb{C}^{*})^{n+1}$.

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  The composition $\pi_{\mathcal{F}}\circ\operatorname{Log}_{t}|_{Q^{n}}$ is a stratified $T^{n}$-fibration that satisfies to all hypotheses of Theorem 3.

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  the closure $\bar{Q^{n}}$ of $Q^{n}$ in ${\mathbb{C}}{\mathbb{P}}^{n+1}\supset(\mathbb{C}^{*})^{n+1}$ is a smooth manifold isotopic to $H$.

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  $Q^{n}$ is invariant with respect to the action of the symmetric group $S_{n+2}$ on $(\mathbb{C}^{*})^{n+1}$ (see above).

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  For a sufficiently large $M>0$

  $$Q^{n}\cap(\mathbb{C}^{*})^{n+1}_{-M}=Q^{n-1}\times\\ C^{*}_{-M},$$

  where $(\mathbb{C}^{*})^{n+1}_{-M}=\{(z_{1},\dots,z_{n+1})\in(\mathbb{C}^{*})^{n+1}\ |\ \log|z_{n+1}|<-M\}$ and $\mathbb{C}^{*}_{-M}=\{z\in\mathbb{C}^{*}\ |\ \log|z|<-M\}$. In particular, the intersection $Q^{n}\cap(\mathbb{C}^{*})^{n+1}_{-M}$ is invariant under a translation $z_{n+1}\mapsto cz_{n+1}$, $0<c<1$.