there exists a symmetry group and the system can be reduced to two degrees of freedom [@Yeh; @KhIISavJPA]. Therefore, for the general case we take $a \ne b$ and choose the notation in such a way that $a > b > 0$. Then the equations of the phase space ${\mathcal{P}}^6$ (the geometric integrals) take the form $$\label{eq2_15}
  z_1^2+x_1 y_2 =r^2, \quad z_2^2+x_2 y_1 =r^2, \quad x_1 x_2 + y_1 y_2 +2 z_1 z_2 = 2 p^2,$$ where $p^2=a^2+b^2$ and $r^2=a^2-b^2$ ($p>r>0$). Since the components of $\mathbf{M}, {{\boldsymbol \alpha}}, {{\boldsymbol \beta}}$ are real, equations define in $\mathbb{C}^9$ the subspace $V^9$. The latter is ensional over ${\mathbb{R}}$. In turn, the phase space ${\mathcal{P}}^6$ is defined in $V^9$ by equations .

Let $f$ be an arbitrary function on $V^9$. For brevity, the term “critical point of $f$” will always mean a critical point of the restriction of $f$ to ${\mathcal{P}}^6$. Similarly, $\rmd f$ means the restriction of the differential of $f$ to the set of vectors tangent to ${\mathcal{P}}^6$.

While calculating critical points of various functions (in the above sense), it is convenient to avoid introducing Lagrange multipliers for the restrictions . Notice that the following vector fields $$\begin{array}{l}
X_1=\partial_{w_1}, \; X_2=\partial_{w_2},\;X_3=\partial_{w_3}, \\
