suppose that $\lambda=0$.

Let us calculate the derivative in virtue of the system of the function $F$ which serves as a partial integral for the extension of the 1st Appelrot class and is defined by equation : $$\label{eq5_1}
  F'=\{H,F\}=\frac{r^2}{2 \sqrt{x_1 x_2}} F_1.$$ Here prime stands for $\rmd/\rmd(\rmi t)$, and with the notation $$F_1= \frac{x_2}{x_1}\xi_1-\frac{x_1}{x_2}\xi_2.$$ Due to on ${\mathcal{M}}_1$ this function vanishes identically but in general we obtain $$\label{eq5_2}
  F'_1=\{H,F_1\}=\frac{1}{\sqrt{x_1 x_2}}\left(\frac{x_2}{x_1}\xi_1+\frac{x_1}{x_2}\xi_2 \right)F.$$ We see that the equations $$\label{eq5_3}
  F=0, \qquad F_1=0$$ form a pair of invariant relations for the GTFG system with $\lambda=0$. For the classical Kowalevski case (${\varepsilon}_0 =1, {\varepsilon}_1 =0, p=r=a, G=L_{\rm class}^2$) this system, obviously, turn into the system with the integral constants satisfying . For the top in a double field the system turns into the system found in [@Kh32].

\[theo2\] Let ${\mathcal{M}}_2$ denote the closure of the set defined by $\eqref{eq5_3}$. Then ${\mathcal{M}}_2$ is a critical subsystem generated by the conditions $\eqref{eq2_12}$ with $$\label{eq5_4}
  {\mathcal{L}}={\mathcal{L}}_2= (p^2h-2g+{\varepsilon}_1 ^2 r^4)^2-r^4 k.$$

The set is well defined for $x_1 x_2 \ne 0$. Then due to , its closure is preserved by the phase flow. It is easy to check the following identity $$\label{eq5_5}
  \left[2(p^2 H -2 G+{\varepsilon}_1 ^2 r^4)-F^2 