Structurally Stable Nondegenerate Singularities of Integrable Systems

Abstract

In this paper, we study singularities of the Lagrangian fibration given by a completely integrable system. We prove that a nondegenerate singular fiber satisfying the so-called connectedness condition is structurally stable under (small enough) real-analytic integrable perturbations of the system. In other words, the topology of the fibration in a neighborhood of such a fiber is preserved after any such perturbation. As an illustration, we show that a simple saddle-saddle singularity of the Kovalevskaya top is structurally stable under real-analytic integrable perturbations and not structurally stable under $C^\infty$-smooth integrable perturbations.

A. Proof of Principal Lemma, Theorem 3, and Corollary 1

Let $\mathscr{L}\subset M$ be a compact rank-$r$ fiber satisfying the connectedness condition (Definition 4), and let $\mathscr{O}_0\subset \mathscr{L}$ be the corresponding nondegenerate rank-$r$ orbit. Suppose Williamson type of $\mathscr{O}_0$ is $(k_e,k_h,k_f)$. By Theorem 2 (b), there exist a neighborhood $U(\mathscr{O}_0)$ of $\mathscr{O}_0$, a local symplectomorphism $\Phi:U(\mathscr{O}_0)\hookrightarrow\mathbb{R}^{2n}$ at $\mathscr{O}_0$ and a real-analytic diffeomorphism $J=(J_1,\dots,J_n):W\hookrightarrow\mathbb{R}^n$ such that

$$ J\circ F\circ\Phi^{-1} = (h_1,\dots,h_n)$$

(A.1)

(cf. (1.1), (1.2)), so $J\circ F$ is a Vey momentum mapping at the orbit $\mathscr{O}_0$ (Definition 3).

Since $\mathscr{O}_0$ is nondegenerate and compact, it is symplectically structurally stable in a strong sense by Theorem 5. Thus, for a perturbed system, we have similar objects $\tilde U(\mathscr{O}_0)$, $\tilde\Phi$ and $\tilde J$ close to $U(\mathscr{O}_0)$, $\Phi$ and $J$ resp., where $\tilde\Phi:\tilde U(\mathscr{O}_0)\to\mathbb{R}^{2n}$ is a perturbed local symplectomorphism, and $\tilde J:\tilde W\to\mathbb{R}^n$ a perturbed diffeomorphism (as in the proof of Theorem 5) near the compact rank-$r$ orbit $\mathscr{O}_0$. In particular, we have a perturbed Vey momentum mapping $\tilde J\circ \tilde F$ near $\mathscr{O}_0$ (Definition 7).

Denote $\mathbb{K}_i\cap \mathscr{L}$ by $K_i$ (for $r+1\le i\le n$). Due to Remark 4, to prove Theorem 3, it suffices to prove the implications (i, ii)$\Longrightarrow$(iii) and ((i), nondegeneracy of $\mathscr{L}$)$\Longrightarrow$(ii). The Principal Lemma and Theorem 3 readily result from the following lemma.

Lemma 1.

Under the above assumptions, the following holds.

(a) The functions

$$J_s\circ F, \quad 1\le s\le r+k_e, \qquad J_{r+k_e+k_h+2j}\circ F, \quad 1\le j\le k_f,$$

(corresponding to regular and elliptic components and elliptic parts of focus-focus components at $\mathscr{O}_0$, cf. (A.1)) generate a Hamiltonian $(S^1)^{r+k_e+k_f}$-action near the fiber $\mathscr{L}$ w.r.t. $\omega$. The fiber $\mathscr{L}$ is fixed under the $(S^1)^{k_e}$-subaction of this action; each $\mathbb{K}_{r+i}$ is a Bott critical points set of the function $J_{r+i}\circ F$, $1\le i\le k_e$. The “perturbed” functions

$$ \tilde J_s\circ\tilde F, \quad 1\le s\le r+k_e, \qquad \tilde J_{r+k_e+k_h+2j}\circ\tilde F, \quad 1\le j\le k_f,$$

(A.2)

generate a Hamiltonian $(S^1)^{r+k_e+k_f}$-action near the fiber $\mathscr{L}$ w.r.t. $\tilde\omega$.

(b) If $K_j$ corresponds to a hyperbolic component $h_j$ in (A.1), then $\mathbb{K}_j$ is a Bott critical points set of the function $J_j\circ F$, the function $iJ_j^\mathbb{C}\circ F^\mathbb{C}$ generates a Hamiltonian $S^1$-action near $K_j$ w.r.t. $\omega^\mathbb{C}$, and the function $i\tilde J_j^\mathbb{C}\circ\tilde F^\mathbb{C}$ generates a Hamiltonian $S^1$-action near $K_j$ w.r.t. $\tilde\omega^\mathbb{C}$.

If $K_j$ corresponds to focus-focus components $h_{j-1},h_j$ in (A.1), then $\mathbb{K}_j$ is a Bott critical point set of each function $J_{j-1}\circ F$ and $J_j\circ F$, the functions $iJ_{j-1}^\mathbb{C}\circ F^\mathbb{C}$ and $J_j^\mathbb{C}\circ F^\mathbb{C}$ generate a Hamiltonian $(S^1)^2$-action near $K_j$ w.r.t. $\omega^\mathbb{C}$, and the functions $i\tilde J_{j-1}^\mathbb{C}\circ\tilde F^\mathbb{C}$ and $\tilde J_j^\mathbb{C}\circ\tilde F^\mathbb{C}$ generate a Hamiltonian $(S^1)^2$-action near $K_j$ w.r.t. $\tilde\omega^\mathbb{C}$.

(c) Suppose that $\mathscr{L}$ is almost nondegenerate (see Subsec. 1.3), and $\mathscr{L}$ is either nondegenerate or satisfies the condition (ii) from Theorem $3.$ Suppose that an orbit $\mathscr{O}\subset \mathscr{L}$ has rank $r'$ and lies in $k_h'$ of the $k_h$ subsets $K_j$ corresponding to hyperbolic components $h_j$ and in $k_f'$ of the $k_f$ subsets $K_j$ corresponding to focus-focus components $h_{j-1},h_j$ in (A.1). Then $r'+k_e+k_h'+2k_f'=n$; $\mathscr{O}$ is nondegenerate of Williamson type $(k_e,k_h',k_f')$ and rank $r'=r+k_h-k_h'+2(k_f-k_f')\ge r$. In particular, the $(S^1)^r$-action generated by $J_1\circ F,\dots,J_r\circ F$ (corresponding to regular components at $\mathscr{O}_0$) is locally free on $\mathscr{O}$ $($and, hence, on $\mathscr{L}).$ Furthermore, the mapping $J\circ F$ is a Vey momentum mapping at $\mathscr{O}$ $($Definition 3$)$ w.r.t. $\omega$ with the same regular, elliptic, hyperbolic and focus-focus components apart from $k_h-k_h'$ hyperbolic and $2(k_f-k_f')$ focus-focus components at $\mathscr{O}_0$ which are regular components at $\mathscr{O}$; the mapping $\tilde J\circ\tilde F$ is a perturbed Vey momentum mapping near $\mathscr{O}$ $($Definition 7$)$ w.r.t. $\tilde\omega$.

Proof.

(a) Observe that the functions (A.2) from the Vey presentation generate a Hamiltonian $(S^1)^{r+k_e+k_f}$-action on $\tilde U(\mathscr{O}_0)$ w.r.t. $\tilde\omega$, moreover the $(S^1)^r$-subaction is locally-free on $\tilde U(\mathscr{O}_0)$, and $\mathscr{O}_0$ is fixed under the unperturbed $(S^1)^{k_e+k_f}$-subaction. Since $\mathscr{L}$ is compact, the time-$2\pi$ mapping of the flow of each vector field $X_{\tilde J_s\circ\tilde F}$, $1\le s\le r+k_e$ or $s=r+k_e+k_h+2j$, $1\le j\le k_f$, is well-defined on a neighborhood of $\mathscr{L}$. Since this mapping is the identity on a $U(\mathscr{O}_0)$, and $\mathscr{L}$ is connected, it follows by uniqueness of analytic continuation that it is the identity on a neighborhood of $\mathscr{L}$. Thus, the functions (A.2) generate a Hamiltonian $(S^1)^{r+k_e+k_f}$-action on some neighborhood of $\mathscr{L}$.

Due to [49, Proposition 2.6], the fiber $\mathscr{L}$ is fixed under the unperturbed $(S^1)^{k_e}$-subaction.

(b) Suppose that $h_j$ is a hyperbolic component in (A.1). It follows from the Vey presentation (1.1), (1.2), (A.1) that the function $iJ_j^\mathbb{C}\circ F^\mathbb{C}$ generates a Hamiltonian $S^1$-action on some neighbourhood $U(\mathscr{O}_0)^\mathbb{C}$ of $\mathscr{O}_0$ in $M^\mathbb{C}$ w.r.t. $\omega$. Similarly, the perturbed function $i\tilde J_j^\mathbb{C}\circ\tilde F^\mathbb{C}$ generates a Hamiltonian $S^1$-action on some neighborhood $\tilde U(\mathscr{O}_0)^\mathbb{C}$ w.r.t. $\tilde\omega$.

By definition of $K_j$, we have $\mathscr{O}_0\subseteq K_j\subseteq \mathscr{L}$ and $\mathrm{d}(J_j\circ F)=0$ at each point of $K_j$. Therefore, $\mathscr{O}_0$ is fixed under the $S^1$-action generated by $iJ_j^\mathbb{C}\circ F^\mathbb{C}$, and the time-$2\pi$ mapping of the flows of the vector fields $X_{iJ_j\circ F}$ and $X_{i\tilde J_j\circ\tilde F}$ are well-defined on some neighborhood of $K_j$ in $M^\mathbb{C}$. Since this time-$2\pi$ mappings are the identity on $U(\mathscr{O}_0)^\mathbb{C}$ and $K_j$ is connected, it follows by uniqueness of analytic continuation that these time-$2\pi$ mappings are the identity on some neighborhood of $K_j$. Thus, the functions $iJ_j\circ F$ and $i\tilde J_j\circ\tilde F$ generate Hamiltonian $S^1$-actions on some neighborhoods $U(K_j)^\mathbb{C}$ and $\tilde U(K_j)^\mathbb{C}$ of $K_j$. We also showed that $(\mathbb{K}_j)^\mathbb{C}$ is the fixed points set on $U(K_j)$ of the unperturbed $S^1$-action, whence $\mathbb{K}_j$ is a symplectic submanifold and it is a Bott critical points set of $J_j\circ F$.

If $h_{j-1},h_j$ are focus-focus components in (A.1), then similar arguments show that

• • the functions $iJ_{j-1}^\mathbb{C}\circ F^\mathbb{C},J_j^\mathbb{C}\circ F^\mathbb{C}$ generate a Hamiltonian $(S^1)^2$-action on some neighborhood $U(K_j)^\mathbb{C}$ of $K_j$ in $M^\mathbb{C}$ w.r.t. $\omega^\mathbb{C}$,

• • $(\mathbb{K}_j)^\mathbb{C}$ is the fixed points set of this $(S^1)^2$-action,

• • the perturbed functions $i\tilde J_{j-1}^\mathbb{C}\circ\tilde F^\mathbb{C}, \tilde J_j^\mathbb{C}\circ\tilde F^\mathbb{C}$ generate a Hamiltonian $(S^1)^2$-action on some neighborhood $\tilde U(K_j)^\mathbb{C}$ of $K_j$ w.r.t. $\tilde\omega^\mathbb{C}$.

(c) By (a), the “regular” and the “elliptic” Vey functions $J_s\circ F$, $1\le s\le r+k_e$, generate a Hamiltonian $(S^1)^{r+k_e}$-action on some neighborhood of $\mathscr{L}$, and the $(S^1)^{k_e}$-subaction is fixed on $\mathscr{L}$.

By (b), we have $k_h'$ “hyperbolic” functions

$$ iJ_{\ell_j}^\mathbb{C}\circ F^\mathbb{C}, \qquad 1\le j\le k_h',$$

(A.3)

and $2k_f'$ “focus-focus” pairs of functions

$$ iJ_{\ell_j-1}^\mathbb{C}\circ F^\mathbb{C}, \quad J_{\ell_j}^\mathbb{C}\circ F^\mathbb{C}, \qquad k_h'+1\le j\le k_h'+k_f',$$

(A.4)

generating a Hamiltonian $(S^1)^{k_h'+2k_f'}$-action on some neighborhood $U(\mathscr{O})^\mathbb{C}$ of $\mathscr{O}$, and this action is fixed on $\mathscr{O}$.

Let us first show that $r'+k_e+k_h'+2k_f'=n$ and $\mathscr{O}$ is nondegenerate of Williamson type $(k_e,k_h',k_f')$. Choose a point $m'\in\mathscr{O}$. Consider two cases.

Case 1: $\mathscr{O}$ is compact. Thus, $k_h'=k_h$ and $k_f'=k_f$ by the connectedness condition. Thus, $m'$ is a fixed point of the Hamiltonian $(S^1)^{n-r}$-action on $U(\mathscr{O})^\mathbb{C}$ generated by the functions $J_{r+i}\circ F$, $1\le i\le k_e$, $iJ_{r+k_e+j}^\mathbb{C}\circ F^\mathbb{C}$, $1\le j\le k_h$, and $iJ_{r+k_e+k_h+2j-1}^\mathbb{C}\circ F^\mathbb{C}, J_{r+k_e+k_h+2j}^\mathbb{C}\circ F^\mathbb{C}$, $1\le j\le k_f$. Therefore $r'=\operatorname{rank}\mathrm{d} F(m')=\operatorname{rank}\mathrm{d}(J\circ F)(m')\le r=\operatorname{rank}\mathrm{d} F(m_0)$. But $m_0$ has minimal rank on $\mathscr{L}$ by connectedness condition. Therefore $r'=r$, thus the functions $J_s\circ F$, $1\le s\le r$, generate a locally-free $(S^1)^r$-action on $\mathscr{O}$.

Thus, $m'$ is a rank-$r$ point of the Hamiltonian $(S^1)^n$-action generated by the above functions (having the form $J_s\circ F$, $iJ_j\circ F$). By [30, Theorem 3.10], there exists a real-analytic symplectomorphism $\Phi':(U(\mathscr{O}),\omega)\to(V/\Gamma',\omega_{can})$ such that

$$J_s\circ F\circ{\Phi'}^{-1}=h_s, \quad 1\le s\le r,$$

where $h_1,\dots,h_r$ are the regular components on $V$, while

$$J_i\circ F\circ{\Phi'}^{-1} = \sum\limits_{j=r+1}^n k_{ij} h_j' , \qquad r+1\le i\le n,$$

for some integers $k_{ij}$ and $n-r$ quadratic functions $h_i'$ of elliptic, hyperbolic and focus-focus types (the number of components $h_i'$ of each type is not necessary $k_e,k_h,k_f$ as in (A.1)). As we showed above, each $\mathbb{K}_i^\mathbb{C}$ ($i>r$) is a fixed point set of the corresponding $S^1$-subaction (resp. $(S^1)^2$-subaction) of the $(S^1)^{n-r}$-action on $U(\mathscr{O})^\mathbb{C}$, and the type of this subaction is given by $h_i$ (resp. $h_{i-1},h_i$) in (A.1).

Since, by assumption, $\mathscr{L}$ either is nondegenerate or satisfies the condition (ii) from Theorem 3, we conclude that

$$J_i\circ F\circ{\Phi'}^{-1}=h_i, \quad r+1\le i\le n,$$

after changing $\Phi'$ if necessarily. In particular, $\mathscr{O}$ is nondegenerate and has Williamson type $(k_e,k_h,k_f)$ and rank $r'=r$, moreover $J\circ F$ is a Vey momentum mapping at $\mathscr{O}$.

Case 2: $\mathscr{O}$ is noncompact. Thus, its closure $\overline{\mathscr{O}}$ contains a compact orbit $\mathscr{O}_1\subset \mathscr{L}$ (because $\mathscr{L}$ is almost nondegenerate). By Case 1, $\mathscr{O}_1$ is nondegenerate of rank $r$ and Williamson type $(k_e,k_h,k_f)$, moreover $\mathscr{O}_1$ lies in each $K_i$, $r+1\le i\le n$, and there exists a real-analytic symplectomorphism $\Phi_1:(U(\mathscr{O}_1),\omega)\to(V/\Gamma_1,\omega_{can})$ such that $J_i\circ F\circ\Phi_1^{-1}=h_i$, $1\le i\le n$.

Since $\mathscr{O}_1$ is nondegenerate and $\mathscr{O}_1\subset\overline{\mathscr{O}}$, we conclude that $\mathscr{O}$ is nondegenerate too, moreover (by Remark 3) it has Williamson type $(k_e,k_h-a,k_f-b)$ and is diffeomorphic to $\mathbb{R}^{a+b}\times(S^1)^{r+b}$, for some $a,b\in\mathbb{Z}_+$. Since $\mathscr{O}_1$ lies in each $K_i$, $r+1\le i\le n$, it follows that $k_h'=k_h-a$ and $k_f'=k_f-b$. Thus, $r'=r+a+2b$ and

$$r'+k_e+k_h'+2k_f'=r+k_e+k_h+2k_f=n,$$

as required. We also obtain that the functions $J_i\circ F$, $1\le i\le r$, generate a locally-free $(S^1)^r$-action on $\mathscr{O}$.

It remains to show that $J\circ F$ is a Vey momentum mapping at $\mathscr{O}$ (Definition 3) w.r.t. $\omega$, and $\tilde J\circ\tilde F$ is a perturbed Vey momentum mapping near $\mathscr{O}$ (Definition 7) w.r.t. $\tilde\omega$. On one hand, by (a) and (b), $\mathscr{O}$ is fixed under the Hamiltonian $(S^1)^{n-r'}$-action on $U(\mathscr{O})^\mathbb{C}$. On the other hand, as we showed above, $\mathscr{O}$ is contained in $k_e+k_h'+k_f'$ subsets $\mathbb{K}_i^\mathbb{C}$, $i\in\{r+1,\dots,r+k_e\}\cup\{\ell_j\}_{j=1}^{k_h'}$ (resp. $i\in\{\ell_j\}_{j=k_h'+1}^{k_h'+k_f'}$), see (A.3), (A.4), each of which is a fixed point set of the corresponding $S^1$-subaction (resp. $(S^1)^2$-subaction) of the $(S^1)^{n-r'}$-action on $U(\mathscr{O})^\mathbb{C}$, and the type of this subaction is given by $h_i$ (resp. $h_{i-1},h_i$) in (A.1).

But, by the assumption, $\mathscr{L}$ is nondegenerate or satisfies the condition (ii) from Theorem 3, therefore the $k_e+k_h'+k_f'$ symplectic submanifolds $\mathbb{K}_i\cap U(\mathscr{O})$ are pairwise transversal and have symplectic pairwise intersections at $m'\in\mathscr{O}$. It follows from Lemma 2 (a) that there exists a real-analytic symplectomorphism $\Phi':(U(m'),\omega)\to(\mathbb{R}^{2n},\omega_{can})$ such that the $n-r'=k_e+k_h'+2k_f'$ functions

$$\begin{array}{l} J_{r+i}\circ F\circ{\Phi'}^{-1}, \quad 1\le i\le k_e,\qquad\qquad J_{\ell_j}\circ F\circ{\Phi'}^{-1}, \quad 1\le j\le k_h', \\ J_{\ell_j-1}\circ F\circ{\Phi'}^{-1},\ J_{\ell_j}\circ F\circ{\Phi'}^{-1}, \qquad k_h'+1\le j\le k_h'+k_f', \end{array}$$

coincide with the quadratic functions $h_i$ of elliptic, hyperbolic and focus-focus types, resp., while the remaining $r'$ functions $J_{i}\circ F\circ{\Phi'}^{-1}$ (whose differentials are automatically linearly independent at $m'$, since $r'=\operatorname{rank}\mathrm{d}(J\circ F)(m')$) are linear functions $\lambda_1,\dots,\lambda_{r'}$.

Thus, $\mathscr{O}$ is nondegenerate of Williamson type $(k_e,k_h',k_f')$ and rank $r'$, and the mapping $J\circ F$ is a Vey momentum mapping at $m'$. In fact, we have even more: it is a Vey momentum mapping at $\mathscr{O}$ (Definition 3), since, due to (a), the remaining $r'=r+k_h-k_h'+2k_f-2k_f'$ functions (namely, the functions $J_s\circ F$, $1\le s\le r$, and the remaining $k_h-k_h'$ hyperbolic functions $J_j\circ F$ and $k_f-k_f'$ focus-focus pairs of functions $J_{j-1}\circ F, J_j\circ F$) generate an $\mathbb{R}^{k_h-k_h'+k_f-k_f'}\times(S^1)^{r+k_f-k_f'}$-action near $\mathscr{L}$, which is locally-free near $\mathscr{O}$, and we can use this action for extending the local symplectomorphism $\Phi'$ to a neighborhood of the cylinder $\mathscr{O}\approx \mathbb{R}^{k_h-k_h'+k_f-k_f'}\times(S^1)^{r+k_f-k_f'}$, as in the proof of Theorem 2 (b).

Due to (a) and (b), the $n-r'=k_e+k_h'+2k_f'$ perturbed functions $\tilde J_{r+i}\circ\tilde F$, $1\le i\le k_e$, $i\tilde J_{\ell_j}\circ\tilde F$, $1\le j\le k_h'$, and $i\tilde J_{\ell_j-1}\circ\tilde F,\tilde J_{\ell_j}\circ\tilde F$, $k_h'+1\le j\le k_h'+k_f'$, generate a “perturbed” Hamiltonian $(S^1)^{n-r'}$-action near $\mathscr{O}$. Since the mapping $\tilde J\circ\tilde F$ is close to $J\circ F$, which is a Vey momentum mapping at $\mathscr{O}$ by above, it follows from Lemma 2 (b) that $\tilde J\circ\tilde F$ is a perturbed Vey momentum mapping near $m'$ (Definition 7). In fact, we have even more: it is a perturbed Vey momentum mapping near $\mathscr{O}$, since we can extend the corresponding local symplectomorphism $\tilde\Phi'$ to a neighborhood of the cylinder $\mathscr{O}\approx \mathbb{R}^{k_h-k_h'+k_f-k_f'}\times(S^1)^{r+k_f-k_f'}$ using the perturbed locally-free $\mathbb{R}^{k_h-k_h'+k_f-k_f'}\times(S^1)^{r+k_f-k_f'}$-action generated by $\tilde J_s\circ\tilde F$, $1\le s\le r$, and the remaining $k_h-k_h'$ hyperbolic functions $\tilde J_j\circ\tilde F$ and $k_f-k_f'$ focus-focus pairs of functions $\tilde J_{j-1}\circ\tilde F, \tilde J_j\circ\tilde F$.

Proof Proof of Corollary 1.

We have to prove the equivalence of four conditions. It follows from Theorem 3 that all of these conditions except for the last one are pairwise equivalent, and the last one implies the previous ones. Moreover the last one follows from the previous one, provided that (iii) implies (v). It is left to note that the latter implication is the Zung topological classification [49, Theorem 7.3].

B. Local normal form and its rigidity

Here we give a proof of Theorem 2 using the following lemma, which we also use (in Sec. 2 and App. A) in the proofs of Theorems 5, 3 and Principal Lemma.

Lemma 2.

Suppose $m_0\in M$ is a singular rank-$r$ point of a real-analytic integrable system $(M,\omega,F)$. Suppose the first differentials of the functions $f_{r+1},\dots,f_n$ at $m_0$ vanish and, in some canonical chart $\Phi_0:(U_0,\omega)\hookrightarrow(\mathbb{R}^{2n},\omega_{can})$ with $\Phi_0(m_0)=0$, the second differentials of $f_{r+1}\circ\Phi_0^{-1},\dots,f_n\circ\Phi_0^{-1}$ at $0$ coincide with the second differentials of $h_{r+1},\dots,h_n$ in (1.1), and $\mathrm{d}(f_s\circ\Phi_0^{-1})(0)=\mathrm{d}\lambda_s$ for $1\le s\le r$. Then the following assertions hold.

(a) There exist a neighborhood $U$ of $m_0$ in $M^\mathbb{C}$ , a neighborhood $W\supseteq F^\mathbb{C}(U)$ of $F(m_0)$ in ${\mathbb C}^n$ , and a unique Hamiltonian $(S^1)^{n-r}$ -action on $U$ generated by the functions

$$ \begin{array}{c} J_{r+1}\circ F,\dots,J_{r+k_e}\circ F, \quad iJ_{r+k_e+1}\circ F,\dots,iJ_{r+k_e+k_h}\circ F, \\ iJ_{r+k_e+k_h+1}\circ F,J_{r+k_e+k_h+2}\circ F,\quad \dots,\quad iJ_{n-1}\circ F,J_n\circ F, \end{array}$$

(B.1)

for some real-analytic functions $J_j:W\to\mathbb{C}$ with $J_j(z)=z_j+O(|z|^2)$ as $z\to0$ , $r+1\le j\le n$ . There exists a real-analytic symplectomorphism $\Phi:(U\cap M,\omega)\hookrightarrow(\mathbb{R}^{2n},\omega_{can})$ such that $\Phi(m_0)=0$ , $\mathrm{d}\Phi(m_0)=\mathrm{d}\Phi_0(m_0)$ , and $J\circ F\circ\Phi^{-1}=(h_1,\dots,h_n)$ , where $J_s(z):=z_s$ for $1\le s\le r$ , $J=(J_1,\dots,J_n):W\to\mathbb{C}^n$ . For a fixed $J$ , any two such symplectomorphisms $\Phi,\Phi'$ near $m_0$ are related by $(\Phi^{-1}\circ\Phi')^2=\phi_{S\circ J\circ F}^1$ , for some real-analytic function $S=S(z_1,\dots,z_n)$ , where $\phi_f^t$ denotes the Hamiltonian flow generated by the function $f$ .

(b) The above $(S^1)^{n-r}$ -action and its normalization are persistent and rigid (resp.) under real-analytic integrable perturbations in the following sense. Suppose we are given a neighborhood $U_1$ of $m_0$ in $M^{\mathbb C}$ and a neighborhood $W_1$ of the origin in ${\mathbb C}^n$ having compact closures $\overline{U_1}\subset U$ and $\overline{W_1}\subset W$ , and an integer $k\in{\mathbb Z}_+$ . Then there exists $\varepsilon>0$ such that, for any (“perturbed”) real-analytic integrable Hamiltonian system $(U\cap M,\tilde\omega,\tilde F)$ whose holomorphic extension to $U$ is $\varepsilon-$ close to $(U,\omega^{\mathbb C},F^{\mathbb C})$ in $C^0$ -norm, the following properties hold. On some neighborhood $\tilde U\supseteq U_1$ , there exists a unique $\tilde F^{\mathbb C}$ -preserving Hamiltonian $($ w.r.t. the “perturbed” symplectic structure $\tilde\omega)$ $(S^1)^{n-r}$ -action generated by functions

$$ \begin{array}{c} \tilde J_{r+1}\circ\tilde F,\dots,\tilde J_{r+k_e}\circ\tilde F, \quad i\tilde J_{r+k_e+1}\circ \tilde F,\dots,i\tilde J_{r+k_e+k_h}\circ \tilde F, \\ i\tilde J_{r+k_e+k_h+1}\circ \tilde F,\tilde J_{r+k_e+k_h+2}\circ \tilde F,\quad \dots,\quad i\tilde J_{n-1}\circ \tilde F,\tilde J_n\circ \tilde F, \end{array}$$

(B.2)

where $\tilde J_j(z_1,\dots,z_n)$ , $r+1\le j\le n$ , are real-analytic functions on some neighborhood $\tilde W\supseteq W_1$ that are $O(\varepsilon)-$ close to $J_j(z_1,\dots,z_n)$ in $C^k$ -norm. There exists a real-analytic symplectomorphism $\tilde\Phi:(\tilde U\cap M,\tilde\omega)\hookrightarrow(\mathbb{R}^{2n},\omega_{can})$ whose holomorphic extension to $\tilde U$ is $O(\varepsilon)-$ close to $\Phi^{\mathbb C}$ in $C^k$ -norm such that $\tilde J\circ\tilde F\circ\tilde\Phi^{-1}=(h_1,\dots,h_n)$ , where $\tilde J_s(z)=z_s$ for $1\le s\le r$ , $\tilde J=(\tilde J_1,\dots,\tilde J_n):\tilde W\to\mathbb{R}^n$ . If the system depends on a local parameter $($ i.e., we have a local family of systems $)$ , moreover its holomorphic extension to $M^{\mathbb C}$ depends smoothly $($ resp., analytically $)$ on that parameter, then $J$ and $\Phi$ can also be chosen to depend smoothly (resp., analytically $)$ on that parameter.

Proof.

(a) We divide the proof into two steps.

Step 1. We can extend the functions $f_1,\dots,f_r$ to a system of local canonical coordinates $\Phi=(\lambda,\varphi,x,y)= (\lambda_1,\varphi_1,\dots,\lambda_r,\varphi_r,x_1,y_1,\dots,x_{n-r},y_{n-r}): U\hookrightarrow\mathbb{R}^{2n}$ on a small neighborhood $U$ of $m_0$ such that $f_s=\lambda_s$ for $1\le s\le r$, $\Phi(m_0)=0$, $\mathrm{d}\Phi(m_0)=\mathrm{d}\Phi(m_0)$ and $\omega|_U=\Phi^*\omega_{can}$ (Darboux coordinates), see (1.2). Consider two cases.

Case 1: $r=0$. Consider the 1-parameter family of “rescaling” coordinate systems $\Phi_\varepsilon=(x',y')$ such that $x=\varepsilon x'$, $y=\varepsilon y'$, where $\varepsilon>0$ is a small parameter. Without loss of generality, we can and will assume that $F(m_0)=0$. By Hadamard’s lemma,

$$f_j\circ\Phi_\varepsilon^{-1}(x',y')=f_j\circ\Phi^{-1}(\varepsilon x',\varepsilon y')=\varepsilon^2 f'_j(x',y',\varepsilon)$$

for some real-analytic functions $f'_j(x',y',\varepsilon)$. Clearly, $\omega=\varepsilon^2\Phi_\varepsilon^*(\mathrm{d} x'\wedge\mathrm{d} y')$ where $\mathrm{d} x'\wedge\mathrm{d} y':=\sum\limits_{j=1}^n\mathrm{d} x_j'\wedge\mathrm{d} y_j'$. Choose a small $\varepsilon_0>0$ such that $\Phi(U)\supseteq B_{0,\varepsilon_0}:=\{w\in\mathbb{R}^{2n}\mid|w|<\varepsilon_0\}$. Denote $U_\varepsilon:=\Phi^{-1}(B_{0,\varepsilon})$ for $0<\varepsilon\le\varepsilon_0$. Thus, the rescaling the diffeomorphism $\Phi_\varepsilon:U_\varepsilon\to B_{0,1}$ transforms the integrable Hamiltonian system

$$ (U_\varepsilon,\varepsilon^{-2}\omega,\varepsilon^{-2}F)$$

(B.3)

to the integrable system

$$ (B_{0,1},\mathrm{d} x'\wedge\mathrm{d} y',F'),$$

(B.4)

where $F\circ\Phi_\varepsilon^{-1}=\varepsilon^2F'(x',y',\varepsilon)$, $F'(x',y',\varepsilon):=(f_1'(x',y',\varepsilon),\dots,f_n'(x',y',\varepsilon))$.

Observe that the “unperturbed” system (i.e., (B.4) with $\varepsilon=0$)

$$ (B_{0,1},\mathrm{d} x'\wedge\mathrm{d} y',F'|_{\varepsilon=0}=(h_1,\dots,h_n))$$

(B.5)

coincides with the linearization of the original system at $m_0$, which has the canonical form by assumption of the lemma. Thus, on a small neighborhood $U=U_\varepsilon$ of $m_0$, our system (B.3) can be viewed as a system (B.4) obtained from the (canonical) “unperturbed” system (B.5) by $O(\varepsilon)$-small integrable perturbation. Using this and [30, Lemma 2.3], one can show that, for each $S^1$-subaction of the above $(S^1)^{n}$-action on $(T_{m_0}M)^\mathbb{C}$ (see Step 1), there exists a point $m_1\subset \mathscr{L}^\mathbb{C}$ satisfying the conditions (i)–(iii) of [30, Theorem 2.2(a)]. By [30, Theorem 2.2(a)], on a small open complexification $U^\mathbb{C}$ of $U=U_\varepsilon$, there exists a $F$-preserving Hamiltonian $(S^1)^{n}$-action generated by some functions (B.1) where $J_j=J_j(z_1,\dots,z_n)$ are real-analytic functions such that $J_j(z_1,\dots,z_n)=z_j+O(|z|^2)$, $1\le j\le n$.Footnote

Indeed: $J(F(\Phi^{-1}(x,y)))=J(F(\Phi_\varepsilon^{-1}(x',y')))= J(\varepsilon^2F'(x',y',\varepsilon))=\varepsilon^2J'(F'(x',y',\varepsilon),\varepsilon)$ for some real-analytic mapping $J'(z,\varepsilon)$ such that $J(\varepsilon^2z)=\varepsilon^2J'(z,\varepsilon)$. Hence the rescaling diffeomorphism $\Phi_\varepsilon^\mathbb{C}$ conjugates the $(S^1)^n$-action $(U^\mathbb{C},\varepsilon^{-2}\omega^\mathbb{C},\varepsilon^{-2}J^\mathbb{C}\circ F^\mathbb{C})$ with the $(S^1)^n$-action $(B_{0,1}^\mathbb{C},(\mathrm{d} x'\wedge\mathrm{d} y')^\mathbb{C},{J'}^\mathbb{C}\circ {F'}^\mathbb{C})$. Hence the linearization of $X_{J_j\circ F}$ at $m_0$ is $\mathrm{d}\Phi_\varepsilon(m_0)$-conjugated with the linearization of $X_{J_j'\circ F'}$ at $0$ for any $\varepsilon>0$. But the quadratic part of $F'$ at $x'=y'=0$ does not depend on $\varepsilon$ and coincides with $F'|_{\varepsilon=0}=(h_1,\dots,h_n)$, see (B.5). This implies that $J_j'(z_1,\dots,z_n)=z_j+O(|z|^2)$ and, hence, $J_j(z_1,\dots,z_n)=z_j+O(|z|^2)$, $1\le j\le n$.

Indeed: $J(F(\Phi^{-1}(x,y)))=J(F(\Phi_\varepsilon^{-1}(x',y')))= J(\varepsilon^2F'(x',y',\varepsilon))=\varepsilon^2J'(F'(x',y',\varepsilon),\varepsilon)$ for some real-analytic mapping $J'(z,\varepsilon)$ such that $J(\varepsilon^2z)=\varepsilon^2J'(z,\varepsilon)$. Hence the rescaling diffeomorphism $\Phi_\varepsilon^\mathbb{C}$ conjugates the $(S^1)^n$-action $(U^\mathbb{C},\varepsilon^{-2}\omega^\mathbb{C},\varepsilon^{-2}J^\mathbb{C}\circ F^\mathbb{C})$ with the $(S^1)^n$-action $(B_{0,1}^\mathbb{C},(\mathrm{d} x'\wedge\mathrm{d} y')^\mathbb{C},{J'}^\mathbb{C}\circ {F'}^\mathbb{C})$. Hence the linearization of $X_{J_j\circ F}$ at $m_0$ is $\mathrm{d}\Phi_\varepsilon(m_0)$-conjugated with the linearization of $X_{J_j'\circ F'}$ at $0$ for any $\varepsilon>0$. But the quadratic part of $F'$ at $x'=y'=0$ does not depend on $\varepsilon$ and coincides with $F'|_{\varepsilon=0}=(h_1,\dots,h_n)$, see (B.5). This implies that $J_j'(z_1,\dots,z_n)=z_j+O(|z|^2)$ and, hence, $J_j(z_1,\dots,z_n)=z_j+O(|z|^2)$, $1\le j\le n$.

By [30, Theorem 3.10(a) or Lemma 6.2(a)], the latter $(S^1)^n$-action is linearizable at $m_0$ ([30, Def. 3.1, 3.7]). In other words, there exists a real-analytic symplectomorphism $\hat\Phi:(U,\omega)\hookrightarrow(\mathbb{R}^{2n},\omega_{can})$ sending the point $m_0$ to the origin, with $\mathrm{d}\hat\Phi(m_0)=\mathrm{d}\Phi(m_0)$, and transforming the momentum mapping $J\circ F$ to a collection of quadratic functions on $V=\Phi(\hat U)$, which does not depend on $\varepsilon$ and, hence, coincides with $(h_1,\dots,h_n)$ from (B.5). Thus, $J$ and $\hat\Phi$ have the required properties.

Case 2: $r>0$. One performs a local Hamiltonian reduction and reduces the problem to an $r$-parameter family of integrable systems with $n-r$ degrees of freedom, with a nondegenerate rank-$0$ point $m_0$. This can be done by the same arguments as in the case of a compact orbit $\mathscr{O}$ (see [38, Sec. 4] or [30, Sec. 7]).

In detail: on a small neighborhood $U_0$ of $m_0$, we can extend the functions $f_1,\dots,f_r$ to a system of local canonical coordinates

$$\Phi_0=(\lambda,\varphi,x,y)= (\lambda_1,\varphi_1,\dots,\lambda_r,\varphi_r,x_1,y_1,\dots,x_{n-r},y_{n-r}): U_0\hookrightarrow\mathbb{R}^{2n}$$

such that $f_s=\lambda_s$ for $1\le s\le r$, $\Phi_0(m_0)=0$, and $\omega|_U=\Phi^*\omega_{can}$ (Darboux coordinates), see (1.2). Take a local disk $P=\{\varphi=0\}$ of dimension $2n-r$ that intersects the local orbit $\mathscr{O}$ through $m_0$ transversally at $m_0$. Then the local disk $\{\lambda_1=\mathrm{const},\dots,\lambda_r=\mathrm{const}\}\cap P$ near $m_0$ has an induced symplectic structure and induced functions $f_{r+1},\dots,f_{n}$ that pairwise Poisson commute.

Applying the case of a rank-$0$ point and parameters $\lambda_1,\dots,\lambda_r$, which is a parametric extension of Case 1 (such an extension is valid due to the parametric extensions [30, Theorems 2.2(b) and 3.10(b) or Lemma 6.2(b)] of [30, Theorems 2.2(a) and 3.10(a) or Lemma 6.2(a)]), we can define an $F$-preserving Hamiltonian $(S^1)^{n-r}$-action on $P^\mathbb{C}$ and local functions $\hat x_1,\hat y_1,\dots,\hat x_{n-r},\hat y_{n-r}$ on $P$, such that they form a local symplectic coordinate system on each local disk $\{\lambda_1=\mathrm{const},\dots,\lambda_r=\mathrm{const}\}\cap P$, with respect to which the Hamiltonian $(S^1)^{n-r}$-action is linear and does not depend on the values of $\lambda_1,\dots,\lambda_r$. Moreover we have $\hat x(m_0)=\hat y(m_0)=0$, the local coordinates $(\hat x,\hat y)$ on the local disk $\{\lambda=0\}\cap P$ have the same linearization at $m_0$ as $(x,y)$. We extend $\hat x_1,\hat y_1,\dots,\hat x_{n-r},\hat y_{n-r}$ to functions on $U$ by making them invariant under the local Hamiltonian flows of $X_{f_1},\dots,X_{f_r}$.

Since $\mathrm{d}\omega=0$, it follows [38, Lemma 4.2] that the symplectic structure $\omega$ on $U$ has the form

$$\omega = \sum_{s=1}^r\mathrm{d} \lambda_s\wedge\mathrm{d}(\varphi_s + g_s) + \sum_{j=1}^{n-r} \mathrm{d}\hat x_j\wedge\mathrm{d}\hat y_j ,$$

for some real-analytic functions $g_s$ on a neighborhood of $m_0$ in $U$, that are invariant under the local Hamiltonian flows of $X_{f_1},\dots,X_{f_r}$.

Define $\hat\lambda_s:=\lambda_s=f_s$, $\hat\varphi_s:=\varphi_s+g_s$, and $J_s(z_1,\dots,z_n)=z_s$ for $1\le s\le r$. Then with respect to the coordinate system $\hat\Phi=(\hat\lambda,\hat\varphi,\hat x,\hat y)$ on $U$, the symplectic form $\omega$ on $U$ has the standard form and the Hamiltonian $(S^1)^{n-r}$-action on $P^\mathbb{C}$ is linear and does not depend on $\lambda$. This implies that $\omega=\hat\Phi^*\omega_{can}$ and the functions

$$J_{r+1}\circ F\circ\hat\Phi^{-1},\dots,J_{r+k_e}\circ F\circ\hat\Phi^{-1}, \ iJ_{r+k_e+1}\circ F\circ\hat\Phi^{-1},\dots,iJ_{r+k_e+k_h}\circ F\circ\hat\Phi^{-1},$$

$$J_{r+k_e+k_h+1}\circ F\circ\hat\Phi^{-1},iJ_{r+k_e+k_h+2}\circ F\circ\hat\Phi^{-1},\dots, J_{n-1}\circ F\circ\hat\Phi^{-1},iJ_{n}\circ F\circ\hat\Phi^{-1}$$

generating this linear Hamiltonian $(S^1)^{n-r}$-action are quadratic functions in $x_j,y_j$ and do not depend on $\lambda$. Clearly, the functions $J_{1}\circ F\circ\hat\Phi^{-1},\dots,J_{n}\circ F\circ\hat\Phi^{-1}$ have the canonical form (1.1), and by construction

$$J_s(z_1,\dots,z_n)=z_s \quad \mbox{for } 1\le s\le r, \qquad J_s(z_1,\dots,z_n)=z_s+O(|z|^2) \quad \mbox{for } r+1\le s\le n.$$

Step 2. It remains to prove the last assertion of (a). We will prove it for $r=0$ (the case $r>0$ can be reduced to the case $r=0$ by a local Hamiltonian reduction, as in Step 1).

Suppose $\Phi,\Phi'$ are two local symplectomorphisms at $m_0$ bringing $J\circ F$ to the canonical form. Then $\Psi:=(\Phi^{-1}\circ\Phi')^2$ is a $F$-preserving real-analytic symplectomorphism of a neighborhood of $m_0$ to $M$ fixing $m_0$ and being homotopic to the identity in the space of $F$-preserving homeomorphisms. Take a regular point $m_1\in \mathscr{L}^\mathbb{C}$ close to $m_0$. Consider the rescaling diffeomorphism $\Phi_\varepsilon:U_\varepsilon\to B_{0,1}$ from Step 1. By Hadamard’s lemma, the mapping $\Phi\circ\Psi\circ\Phi_\varepsilon^{-1}:B_{0,1}\to B_{0,1}$ has the form $\varepsilon\Psi_\varepsilon$, where $\Psi_\varepsilon:B_{0,1}\to B_{0,1}$ is a 1-parameter family of real-analytic mappings in $x',y',\varepsilon$. Clearly, $\Psi_\varepsilon$ preserves $J'(F'(x',y',\varepsilon),\varepsilon)$ and $\mathrm{d} x'\wedge\mathrm{d} y'$. One checks that the unperturbed mapping $\Psi_0$ is linear and coincides with $\mathrm{d}(\Phi\circ\Psi\circ\Phi^{-1})(m_0)$. Take a point $m'=(x',y')\in B_{0,1}^\mathbb{C}$ which is a regular point of the singular fiber of the unperturbed system (B.5). Without loss of generality, $m'$ is fixed under the unperturbed linear mapping $\Psi_0$ (this can be achieved by replacing $\Psi$ with its composition with the time-$1$ mapping of the Hamiltonian flow generated by a linear combination of $J_j\circ F$, $1\le j\le n$). We can extend the functions $\lambda_j={J_j'}^\mathbb{C}\circ{F'}^\mathbb{C}$ to a local system of canonical holomorphic coordinates $\lambda_j,\mu_j$ near $m'$ (Darboux coordinates) depending analytically on $\varepsilon$ such that $\mu_j(m')=0$. Since $m'$ is fixed under the unperturbed mapping $\Psi_0$, it follows that, in these coordinates the perturbed mapping $\Psi_\varepsilon^\mathbb{C}$ on a neighborhood $\tilde U(m')^\mathbb{C}$ of the point $m'$ has the form $(\lambda,\mu)\mapsto(\lambda,\mu+\frac{\partial S_\varepsilon^\mathbb{C}}{\partial\lambda})$, for some real-analytic function $S_\varepsilon=S_\varepsilon(\lambda)$ on a neighborhood of the origin such that $S_0(0)=0$ and $\frac{\partial S_0(0)}{\partial\lambda_j}=0$. Thus, on $\tilde U(m')^\mathbb{C}$, the mapping $\Psi_\varepsilon^\mathbb{C}$ coincides with the time-1 mapping of the Hamiltonian flow generated by $S_\varepsilon^\mathbb{C}({J_j'}^\mathbb{C}({F'}^\mathbb{C}(x',y',\varepsilon),\varepsilon))$. Choose a point $m_\varepsilon'=(x_\varepsilon',y_\varepsilon')\in\tilde U(m')^\mathbb{C}$ with $\lambda_j(m')=\mu_j(m')=0$. Thus, $\Psi^\mathbb{C}$ coincides with the time-1 mapping of the Hamiltonian flow generated by $S\circ J\circ F$, where $S(z)=\varepsilon^2S_\varepsilon(z/\varepsilon^{2})$ on a small neighborhood of the point $m_\varepsilon:=\Phi^{-1}(\varepsilon m_\varepsilon')\in \mathscr{L}^\mathbb{C}$. Since the analytic symplectomorphisms $\Psi^\mathbb{C}$ and $(\phi_{S\circ J\circ F}^1)^\mathbb{C}$ are well-defined on some neighborhood $U$ of $m_0$ in $M^\mathbb{C}$ and coincide with each other on a neighborhood of the path $m_u\in \mathscr{L}^\mathbb{C}\cap U$, $0\le u\le\varepsilon$, by analytic continuation they must coincide on the whole $U$.

This yields Lemma 2 (a).

(b) On a neighborhood $\tilde U$ of $m_0$ close to $U$, we can extend the “perturbed” functions $\tilde f_1,\dots,\tilde f_r$ to a “perturbed” system of local canonical coordinates $\tilde \Phi=(\tilde \lambda,\tilde \varphi,\tilde x,\tilde y):\tilde U\hookrightarrow\mathbb{R}^{2n}$ such that $\tilde f_s=\tilde \lambda_s$ for $1\le s\le r$ and $\tilde\omega|_{\tilde U}=\tilde \Phi^*\omega_{can}$ (perturbed Darboux coordinates), see (1.2).

We obtain a “perturbed” $r$-parameter family of integrable systems with $n-r$ degrees of freedom, with parameters $\tilde\lambda_1,\dots,\tilde\lambda_r$. Since the “unperturbed” system with zero values of the parameters ($\lambda_1=\dots=\lambda_r=0$) admits a nondegenerate rank-$0$ point $m_0$, we can derive the assertion (b) from Case 1 of (a) similarly to deriving Case 2 of (a), by applying to the “perturbed” system the “perturbative” extension [30, Theorem 2.2(b) and 3.10(b) or Lemma 6.2(b)] of [30, Theorem 2.2(a) and 3.10(a) or Lemma 6.2(a)].

This yields Lemma 2 (b).

B.1. Proof of Theorem 2

(a) We want to bring our system to a canonical form (1.1), (1.2) on a small neighborhood $U$ of the point $m_0$. This can be done using [44]. Let us give another proof based on Lemma 2 (a) (which we proved using [30]).

After replacing $f_1,\dots,f_n$ by their linear combinations, we can assume that $\mathrm{d} f_j(m_0)=0$ for each $j>r$. In particular, $\mathrm{d} f_1\wedge\dots\wedge\mathrm{d} f_r\ne0$ at $m_0$. Suppose also that $F(m_0)=0$ for $j>r$, which can be achieved by adding a constant to each $f_j$.

On a small neighborhood $U$ of $m_0$, we can extend the functions $f_1,\dots,f_r$ to a system of local canonical coordinates $\Phi_0=(\lambda,\varphi,x,y)= (\lambda_1,\varphi_1,\dots,\lambda_r,\varphi_r,x_1,y_1,\dots,x_{n-r},y_{n-r}): U\hookrightarrow\mathbb{R}^{2n}$ such that $f_s=\lambda_s$ for $1\le s\le r$, $\Phi_0(m_0)=0$, and $\omega|_U=\Phi_0^*\omega_{can}$ (Darboux coordinates), see (1.2).

It follows from the Williamson theorem that (after replacing $f_{r+1},\dots,f_n$ by their linear combinations, and applying to $x,y$ a linear canonical transformation if necessary) the second differentials of $f_{r+1}\circ\Phi_0^{-1}|_{\lambda=0},\dots,f_n\circ\Phi_0^{-1}|_{\lambda=0}$ at the origin have a canonical form, i.e., coincide with the second differentials of $h_{r+1}|_{\lambda=0},\dots,h_n|_{\lambda=0}$ in (1.1). In particular, the linearizations at the point $m_0$ of the restrictions of the Hamiltonian vector fields generated by (B.1) to $\{\lambda=0\}$ have $2\pi$-periodic flows on $(T_{m_0}M)^\mathbb{C}$.

Due to Lemma 2 (a), there exist $J$ and $\Phi$ with the required properties.

(b) Suppose $\mathscr{O}$ is a rank-$r$ orbit, $m_0\in\mathscr{O}$. Since the flows of all $X_{f_i}$ are complete on $\mathscr{O}$, it is diffeomorphic to a cylinder $\mathbb{R}^{r_o}\times(S^1)^{r_c}$, where $r_o$ and $r_c$ are degree of openness and degree of closedness of $\mathscr{O}$, respectively [49, Def. 3.4], $r=r_o+r_c$.

By [19] or [51, 55] (or [49, Theorem 6.1] in the $C^\infty$ case with a proper $F$), there exists a locally-free $F$-preserving Hamiltonian $(S^1)^{r_c}$-action on a neighborhood $U(\mathscr{O})$ of $\mathscr{O}$. This $(S^1)^{r_c}$-action is generated by functions of the form $J_{r_o+1}\circ F,\dots,J_r\circ F$ for some real-analytic functions $J_s(z_1,\dots,z_n)$, $r_o+1\le s\le r$. Without loss of generality, $\partial(J_{r_o+1},\dots,J_r)/\partial(z_{r_o+1},\dots,z_r)\ne0$ and $\mathrm{d} f_1\wedge\dots\wedge\mathrm{d} f_r\ne0$ at some (and hence each) point of $\mathscr{O}$. Without loss of generality, this $(S^1)^{r_c}$-action is effective.

Besides, we can extend to $U(\mathscr{O})^\mathbb{C}$ the Hamiltonian $(S^1)^{n-r}$-action on $U^\mathbb{C}$ generated by $J_j^\mathbb{C}\circ F^\mathbb{C}$, $r+1\le j\le n$, constructed in (a). The above $(S^1)^{r_c}$-action and $(S^1)^{n-r}$-action give rise to the Hamiltonian $(S^1)^{n-r_o}$-action on $U(\mathscr{O})^\mathbb{C}$ generated by $J_j^\mathbb{C}\circ F^\mathbb{C}$, $r_o+1\le j\le n$. Put $J_s(z_1,\dots,z_n):=z_s$, $1\le s\le r_o$. Consider two cases.

Case 1: $r_o=0$, thus the orbit $\mathscr{O}$ is compact. By [30, Theorem 3.10(a)], the above $(S^1)^{n-r_o}$-action is symplectomorphic to a linear model, thus the system $(U(\mathscr{O}),\omega,J\circ F)$ is symplectomorphic to a linear model $(V/\Gamma,\omega_{can},(h_1,\dots,h_n))$ [30, Def. 3.7] having the form (1.1), (1.2). In the terminology of [30, Def. 3.7], this means that the integer $(n-r)\times(n-r)$-matrix $\|p_{j\ell}\|$ (whose columns are “extended” elliptic and hyperbolic resonances of the singularity) is the unity matrix: $p_{j\ell}=\delta_{j\ell}$ (we can achieve this, since our matrix $\|p_{j\ell}\|$ is a nondegenerate square matrix, and we are allowed to replace the functions $J_j\circ F$ by their linear combinations forming a nondegenerate matrix). We can manage that the action of $\Gamma$ is trivial on each elliptic disk $D^2$ and on each focus-focus polydisk $D^2\times D^2$, because the twisting resonances are well-defined only up to adding any linear combinations of the “extended” elliptic resonances [30, Remark 3.11(C)]. The action of $\Gamma$ on $(D^2)^{n-r}$ is effective, since otherwise the above $(S^1)^{r_c}$-action is noneffective.

Case 2: $r_o>0$. We deduce this case from a parametric version of Case 1 (similarly to the proof of Lemma 2 (a), Step 1, Case 2) by considering the corresponding reduced integrable Hamiltonian system with $n-r_o$ degrees of freedom (obtained by local symplectic reduction under the local Hamiltonian action generated by $f_1,\dots,f_{r_o}$). In this way, we see from Case 1 and [30, Theorem 3.10(b)] that the system $(U(\mathscr{O}),\omega,J\circ F)$ is symplectomorphic to a neighborhood of the cylinder $\{0\}^r\times\mathbb{R}^{r_o}\times(S^1)^{r_c}\times\{(0,0)\}^{n-r}$ in the linear model $(V/\Gamma,\omega_{can},(h_1,\dots,h_n))$ having the form (1.1), (1.2), as required.

This yields Theorem 2. □