are derived from the condition that the algebraic curve of the Lax pair has a singular point. For two subsystems the invariant relations in the phase space are found. Another two are defined implicitly by the equations of the type . This form of description does not provide a clear way to calculate the types of corresponding critical points. In what follows, we give explicit equations of three critical subsystems generalizing the Appelrot classes of motion.

To present formulas in a shorter form, let us introduce the linear change of variables ($\rmi^2=-1$) $$\label{eq2_14}
\begin{array}{c}
\begin{array}{ll}
x_1 = (\alpha_1  - \beta_2) + \rmi (\alpha_2  + \beta_1),&
x_2 = (\alpha_1  - \beta_2) - \rmi(\alpha_2  + \beta_1 ), \\
y_1 = (\alpha_1  + \beta_2) + \rmi(\alpha_2  - \beta_1), & y_2 =
(\alpha_1  + \beta_2) -
\rmi(\alpha_2  - \beta_1), \\
z_1 = \alpha_3  + \rmi\beta_3, &
z_2 = \alpha_3  - \rmi\beta_3,
\end{array}\\
w_1 = \frac{1}{2}(M_1  + \rmi M_2) , \quad w_2 = \frac{1}{2}(M_1  - \rmi M_2), \quad w_3 = M_3-\lambda.
\end{array}$$

For convenience, we use the Poisson brackets obtained from the initial Lie–Poisson brackets dividing by $\rmi$. Then for the phase vector $\mathbf{u} = (w_1,w_2,w_3,x_1,x_2,y_1,y_2,z_1,z_2)$ the matrix of the basic brackets $\{u_i,u_j\}$ is $$\left(\begin{array}{ccccccccc}
  