## 3 Characterization of Invariant Relative Orbits
### Siegel-Moser Canonical Transformation  
The Hamilton Equation (3) obtained from \(\tilde{H}\) corresponds to the following linear dynamical system:  
\[\left[ \begin{array}{c}\dot{\mathbf{Q}}\\ \dot{\mathbf{P}}\end{array} \right]=A\left[ \begin{array}{c}\mathbf{Q}\\ \mathbf{P}\end{array} \right] \tag{24}\]  
where  
\[A=\left[ \begin{array}{cccccc}0&1&0&1&0&0\\ -1&0&0&0&1&0\\ 0&0&0&0&0&1\\ 2(1-K_{11})&0&0&0&1&0\\ 0&-1-2K_{22}&0&-1&0&0\\ 0&0&-1-2K_{33}&0&0&0\end{array} \right] \tag{25}\]  
Examining Equation (23), it is possible to state that the system admits an equilibrium point \(\mathbf{x}^{*}=\left[ \begin{array}{cc}\mathbf{Q}=\mathbf{0}&\mathbf{P}= \mathbf{0}\end{array} \right]^{T}\), with one pair of real \((\pm\alpha)\) and two pairs of imaginary \((\pm i\beta_{1},\pm i\beta_{2})\) eigenvalues, corresponding to an equilibrium of the saddle-center-center type. According to the Morse-Palais lemma, in the neighborhood of \(\mathbf{x}^{*}\), the Hamiltonian function can be rearranged as the sum of three quadratic terms: one hyperbola (saddle) and two ellipses (centers). A transformation \(\mathcal{T}_{7}:(\mathbf{Q},\mathbf{P})\in\mathbb{R}\rightarrow(\mathbf{w}, \mathbf{z})\in\mathbb{C}\) is developed here that produces the above-mentioned form, diagonalizes \(A\), and is symplectic (i.e., \(\mathcal{T}_{7}\) is canonical) [32]. Indicating with \(\mathbf{\Lambda}=diag(\alpha,-\alpha,j\beta_{1},-j\beta_{1},j\beta_{2},-j\beta _{2})\) the diagonal matrix of eigenvalues and with \(\mathbf{T}_{\Lambda}\) the matrix whose columns are the corresponding eigenvectors, the following permutation is introduced:\[P=\left( \begin{array}{cccccc}1&2&3&4&5&6\\ 1&4&2&5&3&6\end{array} \right)\]  
to obtain the Siegel-Moser form  
\[\mathbf{T}=P(\mathbf{T}_{\Lambda})=diag(\alpha,j\beta_{1},j\beta_{2},-\alpha,- j\beta_{1},-j\beta_{2}) \tag{26}\]  
For the transformation \(\mathcal{T}_{7}\) to be canonical, \(\mathbf{T}\) shall be transformed into a symplectic form \(\mathbf{S}\). A matrix \(\mathbf{Q}\) is introduced such that  
\[\mathbf{S}=\mathbf{T}\mathbf{Q}^{-1} \tag{27}\]  
having selected  
\[\mathbf{Q}=\left[ \begin{array}{cc}\mathbf{B}_{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}\end{array} \right]\]  
\[\mathbf{J}=\left[ \begin{array}{cc}\mathbf{0}&\mathbf{I}\\ -\mathbf{I}&\mathbf{0}\end{array} \right]\]  
\[\mathbf{B}=\left(\mathbf{J}^{-1}\mathbf{T}^{T}\mathbf{J}\right)^{-1}\]  
where \(\mathbf{I}\) is a \(3\times 3\) identity matrix and \(\mathbf{B}_{1}\) is the \(3\times 3\) matrix corresponding to the first three rows and columns of \(\mathbf{B}\). Finally, to verify that reality condition on the state variables \((\mathbf{Q},\mathbf{P})\), matrix \(\mathbf{S}\) shall be post-multiplied by  
\[\mathbf{R}=diag(1,k_{1},k_{2},1,1/k_{1},1/k_{2})\]  
where \(k_{1}=\mathbf{s}_{2}{}^{T}\mathbf{s}_{5}\), \(k_{2}=\mathbf{s}_{3}{}^{T}\mathbf{s}_{6}\), and \(\mathbf{s}_{i}\) indicate the i-th column of \(\mathbf{S}\). We collect all the results of the transformations above in the following matrix:  
\[\mathbf{T}_{N}=P(\mathbf{T}_{\Lambda})\mathbf{Q}^{-1}\mathbf{R} \tag{28}\]  
which produces  
\[\mathcal{T}_{7}:=\left[ \begin{array}{cc}\mathbf{Q}\\ \mathbf{P}\end{array} \right]=\mathbf{T}_{N}\left[ \begin{array}{cc}\mathbf{w }\\ \mathbf{z}\end{array} \right] \tag{29}\]  
\[\mathbf{T}_{N}=\left[ \begin{array}{cccccc}2c_{1}&c_{1}&0&-2c_{2}&jc_{1}&0\\ c_{3}&2 jc_{1}&0&-c_{3}&2c_{1}&0\\ 0&0&-c_{1}&0&0&-jc_{1}\\ -c_{3}&-jc_{1}&0&c_{1}&-c_{1}&0\\ -c_{1}&-c_{1}&0&c_{2}&-jc_{1}&0\\ 0&0&-jc_{1}&0&0&-c_{1}\end{array} \right] \tag{30}\]  
where the value of the coefficients \(c_{i}\in\mathbb{R}^{+}\) of \(\mathbf{T}_{N}\) can be obtained from Equation (28) and depends on that of the coefficients \(K_{ij}\).