three subsystems with two degrees of freedom. For the gyrostat, one of the subsystems (the Bogoyavlensky case) splits, the parts of the new manifold join the other subsystems. In two of three critical subsystems for the top in a double field the algebraic separation of variables was obtained [@KhSavUMBeng; @KhRCD09]. This turns the topological investigation into a strictly analytical procedure [@KhND11].

Still, further generalization of the problem was found in [@SokTsiE]. To describe this case take the system with the Kowalevski configuration and put $$\displaystyle H^{(0)} = - {\varepsilon}_0 ({\mathbf{e}}_1  {\bdot} {{\boldsymbol \alpha}}+ {\mathbf{e}}_2  {\bdot} {{\boldsymbol \beta}})+\frac{\lambda^2}{2}, \quad   H^{(1)} = - \lambda M_3 - {\varepsilon}_1  \mathbf{M} {\bdot} ({\mathbf{e}}_1  {\times} {{\boldsymbol \alpha}}+ {\mathbf{e}}_2  {\times} {{\boldsymbol \beta}}).$$ The full Hamilton function takes the form $$\label{eq2_13}
\begin{array}{l}
  \displaystyle H = \frac{1}{4}(M_1^2+M_2^2)+\frac{1}{2}(M_3-\lambda)^2 \\[3mm]
 \displaystyle \quad - {\varepsilon}_1  [ (\alpha_2 M_3 - \alpha_3 M_2 ) + (\beta_3 M_1 - \beta_1 M_3 )] -{\varepsilon}_0 (\alpha_1+\beta_2).
\end{array}$$ For this function, the Lax representation with a spectral parameter was given in [@SokTsiE] proving the complete Liouville integrability. The authors of [@SokTsiE] called their case the generalized two fields gyrostat. In what follows we call the system with the Hamilton function the GTFG system. P.E.Ryabov in [@RyabTMF] started the 