$\ell={\rm const}$ is shown in Figure \[fig1\]. We see the reconstruction of the Appelrot classes to form new critical subsystems. Therefore, in this section for the general case we suppose that $\lambda=0$.

\[theo1\] The GTFG system with $\lambda=0$ has the invariant manifold ${\mathcal{M}}_1$ defined by the system of invariant relations $$\label{eq4_1}
  Z_1=0, \qquad Z_2=0,$$ where $$\begin{array}{l}
  Z_1= \frac{1}{4}(M_1^2-M_2^2)+{\varepsilon}_0 (\alpha_1-\beta_2)+{\varepsilon}_1 [({{\boldsymbol \alpha}}{\times}{\mathbf{M}}){\bdot}\mathbf{e}_1-({{\boldsymbol \beta}}{\times}{\mathbf{M}}){\bdot}\mathbf{e}_2] \\[3mm]
  \phantom{ Z_1} - {\varepsilon}_1 ^2({{\boldsymbol \alpha}}^2-{{\boldsymbol \beta}}^2),     \\[3mm]
  Z_2=\frac{1}{2}M_1 M_2+{\varepsilon}_0 (\alpha_2+\beta_1)+{\varepsilon}_1 [({{\boldsymbol \alpha}}{\times}{\mathbf{M}}){\bdot}\mathbf{e}_2+({{\boldsymbol \beta}}{\times}{\mathbf{M}}){\bdot}\mathbf{e}_1-2{\varepsilon}_1  {{\boldsymbol \alpha}}{\bdot}{{\boldsymbol \beta}}].
\end{array}$$ The induced system on ${\mathcal{M}}_1$ is the critical subsystem generated by the conditions $\eqref{eq2_12}$ with $$\label{eq4_2}
  {\mathcal{L}}={\mathcal{L}}_1= k.$$

The proof follows immediately from the fact that in this case $$\label{eq4_3}
  K=Z_1^2+Z_2^2.$$ Obviously, for the classical Kowalevski case we come to equations , , thus obtaining the generalization of the first Appelrot class of motions. If ${{\boldsymbol \beta}}\ne 0$, putting ${\varepsilon}_0 =1, {\varepsilon}_1 =0$ we come to the partial integrable case found by Bogoyavlensky [@Bogo].

Let us point out some features of the critical subsystem ${\mathcal{M}}_1$.

\[prop1\] The set ${\mathcal{M}}_1$ is a smooth 4-dimensional manifold except for the values of the parameters satisfying one of the relations $$\label{eq4_4}
  {\varepsilon}_0 =(a \pm b) {\varepsilon}_1 ^2,$$ when ${\mathcal{M}}_1$ fails to be smooth 