manifolds (critical subsystems with two degrees of freedom). For each subsystem we point out the invariant relations defining its manifold and give a pair of independent first integrals. In terms of these integrals, using the invariant relations we explicitly calculate the types of the corresponding critical points of the integral map. It gives an analytical basis for the global topological study of the whole system with three degrees of freedom.

Preliminaries {#sec0}
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Let $\mathbf{v}$ denote the Lie algebra $e(3,2)=so(3)\oplus({\mathbb{R}}^3 \dot{+}{\mathbb{R}}^3)$, the semi-direct sum of $so(3)$ and two copies of ${\mathbb{R}}^3$. The Euler equations on the adjacent space $\mathbf{v}^*\cong{\mathbb{R}}^9(\mathbf{M},{{\boldsymbol \alpha}},{{\boldsymbol \beta}})$ are (see [@Bogo]) $$\label{eq2_1}
\begin{array}{l}
\displaystyle \frac{\rmd {\mathbf{M}}}{\rmd t} =\mathbf{M} {\times} \frac{\partial H} {\partial \mathbf{M}}  +  {{\boldsymbol \alpha}}{\times} \frac{\partial H} {\partial {{\boldsymbol \alpha}}}+ {{\boldsymbol \beta}}{\times} \frac{\partial H} {\partial {{\boldsymbol \beta}}},\\[3mm]
\displaystyle \frac {\rmd {{\boldsymbol \alpha}}}{\rmd t} = {{\boldsymbol \alpha}}{\times} \frac{\partial H} {\partial \mathbf{M}}, \qquad \frac {\rmd {{\boldsymbol \beta}}}{\rmd t}=
{{\boldsymbol \beta}}{\times} \frac{\partial H} {\partial \mathbf{M}}.
\end{array}$$ These equations have a Hamiltonian form (with the Hamilton function $H$) with respect to the Lie–Poisson brackets on $\mathbf{v}^*$, i.e., for any function $f$ on $\mathbf{v}^*$ we have ${\rmd f}/{\rmd t}=\{H,f\}$. If $$\label{eq2_2}
  H=H^{(2)}+H^{(1)}+H^{(0)},$$ where $H^{(i)}$ is a homogeneous degree $i$ function of 