following two equations $$\label{eq3_2}
  \omega_1^2-\omega_2^2+\alpha_1=0, \qquad 2\omega_1 \omega_2+\alpha_2 =0 .$$

The 2nd and the 3rd classes are generated by the relation $$\label{eq3_3}
  (2\ell^2-a^2h)^2-a^4 k =0.$$ The difference between two classes is in the sign of $2\ell^2-a^2h$. This value is positive in the 2nd class and negative in the 3rd one. Zero value, obviously, gives a subset of motions from the 1st class treated usually as common motions for all three classes. In ${\mathcal{P}}^5$ we have a connected manifold defined by two invariant relations (e.g. [@Kh32]) $$\label{eq3_4}
\begin{array}{c}
\displaystyle{(\alpha _1^2  + \alpha _2^2 )\omega _3 - 2(\alpha _1
\omega _1  + \alpha _2 \omega _2 )\alpha _3  =
0,} \\
\displaystyle{2\alpha _1 \alpha _2 (\omega _1^2  - \omega _2^2  +
\alpha _1 ) - (\alpha _1^2  - \alpha _2^2 )(2\omega _1 \omega _2  +
\alpha _2 ) = 0}.
\end{array}$$

The 4th Appelrot class is generated by the zero discriminant of the last multiplier in $S_{\rm Kow}$ $$27\ell^4-2h(9a^2+h^2-9k)\ell^2+(a^2-k)(a^2+h^2-k)=0.$$ It is more convenient to write the equations of this surface taking $h$ and $s$ for the parameters: $$\label{eq3_5}
k=a^2+(h-s)(h-3s), \qquad \ell^2=s^2(h-s).$$ Appelrot shows that all especially remarkable motions of the 4th class satisfy the following conditions $$\label{eq3_6}
s = -\frac{\ell}{\omega_1}={\rm const}, \qquad 