Y_1= z_2 \partial_{x_2} + z_1 \partial_{y_2} -\frac{1}{2} x_1 \partial_{z_1} - \frac{1}{2} y_1 \partial_{z_2},\\
Y_2= z_1 \partial_{x_1} + z_2 \partial_{y_1}  - \frac{1}{2} y_2 \partial_{z_1}-\frac{1}{2} x_2 \partial_{z_2},\\
Y_3= x_1 \partial_{x_1} - x_2 \partial_{x_2} + y_1 \partial_{y_1} - y_2 \partial_{y_2}
\end{array}$$ are tangent to ${\mathcal{P}}^6 \subset V^9$ and linearly independent at any point of ${\mathcal{P}}^6$. Then the set of critical points of $f$ is defined by the system of equations $$\label{eq2_16}
\begin{array}{lll}
X_1 f=0, & X_2 f=0, & X_3 f=0, \\
Y_1 f=0, & Y_2 f=0, & Y_3 f=0.
\end{array}$$ This fact will be used below.

The Appelrot classes {#sec2}
====================

Consider the classical Kowalevski problem. According to the separated equations obtained by S.Kowalevski [@Kowa] and the constant shift in the variables introduced later by F.Kötter [@Kott], the especially remarkable motions in Appelrot’s definition [@Appel] are possible only if the polynomial in $s$ $$S_{\rm Kow} (s)=[(s-h)^2 - k]\{s[(s-h)^2+a^2-k]-2\ell^2\}$$ has a multiple root. Here $h,\ell,k$ stand for the constants of the integrals – respectively. Appelrot points out the following classes of such motions.

The 1st class is characterized by the relation $$\label{eq3_1}
  k=0.$$ It immediately follows from that the corresponding invariant manifold is defined in the reduced phase space ${\mathcal{P}}^5={\mathbb{R}}^3({{\boldsymbol \omega}}){\times}S^2({{\boldsymbol \alpha}})$ by the 