\end{array}$$ Thus, the system with $H=H_\lambda$, where $$\label{eq2_9}
  H_\lambda = \frac{1}{4}(M_1^2+M_2^2)+\frac{1}{2}M_3^2 - \lambda M_3 -\alpha_1-\beta_2 + \frac{\lambda^2}{2},$$ having three first integrals $H_\lambda, K_{\rm Y}, G_{\rm RS}$ in involution is a Liouville integrable system with three degrees of freedom. It is known as the Kowalevski *gyrostat* in a double field. The relation between $\mathbf{M}$ and ${{\boldsymbol \omega}}$ then is $$M_1=2 \omega_1, \quad M_2=2 \omega_2, \quad M_3=\omega_3+\lambda.$$ The constant term $\frac{\lambda^2}{2}$ in $H_\lambda^{(0)}$ is introduced to make $H_\lambda$ the total energy of the gyrostat. If $\lambda=0$ the system is called the Kowalevski *top* in a double field. Obviously, for ${{\boldsymbol \beta}}=0$ we have $G_{\rm RS}=L_{\rm Y}^2$, so the new integral generalizes the area type integral for a double field.

The Kowalevski top and the Kowalevski gyrostat in a double field up to this moment have not been reduced to quadratures. However, the phase topology of the Kowalevski top in a double field is completely investigated. The collection of results and the relevant references can be found in [@RyabKhMatsbEn; @KhRyabDME]. The Kowalevski gyrostat in a double field is also actively studied [@KhHMJ; @KhIISmir40]. These investigations are based on the notion of a critical subsystem, i.e., an even-dimensional invariant submanifold in the phase space 