For small enough values of ${\varepsilon}_1 $ the integral $N$ does not vanish on ${\mathcal{M}}_1$. The 2-form induced on ${\mathcal{M}}_1$ by the symplectic structure of ${\mathcal{P}}^6$ is non-degenerate everywhere except for the set $$\label{eq4_12}
  N \, F=0.$$

Expression is obtained by a straightforward calculation using . We see that if ${\varepsilon}_1 =0, {\varepsilon}_0 >0$, then $N={\varepsilon}_0 \ne 0$. For ${\varepsilon}_1 \ne 0$ the equation $N=0$ yields $$\begin{array}{c}
  \displaystyle x_1=x_2=0, \quad y_1=\frac{{\varepsilon}_0 +\rmi {\varepsilon}_1  w_3}{{\varepsilon}_1 ^2},\quad y_1=\frac{{\varepsilon}_0 +\rmi {\varepsilon}_1  w_3}{{\varepsilon}_1 ^2},\quad y_2=\frac{{\varepsilon}_0 -\rmi {\varepsilon}_1  w_3}{{\varepsilon}_1 ^2}, \\
  z_1^2=z_2^2=r^2, \quad w_1 =-\rmi {\varepsilon}_1  z_1, \quad w_2=\rmi {\varepsilon}_1  z_2.
\end{array}$$ Then from we obtain ${{\varepsilon}_0 ^2}-2(p^2-r^2){{\varepsilon}_1 ^4}+{w_3^2} {{\varepsilon}_1 ^2}=0$. Therefore, $N$ can vanish only for the values $${\varepsilon}_1 ^4 \geqslant \frac{{\varepsilon}_0 ^2}{2(p^2-r^2)}.$$ The property of the induced 2-form follows from Lemma \[lem1\].

The extension of the 2nd and 3rd Appelrot classes {#sec4}
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As one can see from Figure \[fig1\], the 2nd and 3rd Appelrot classes forming one invariant manifold in the classical Kowalevski case split and glue with other sets of critical points when $\lambda \ne 0$. For the gyrostat in a double field (${\varepsilon}_1 =0$) new subsystems are described in [@KhHMJ]. In this section we take ${\varepsilon}_1 \ne 0$, but still 