topological analysis of the GTFG system. First of all, he obtains the explicit formulas of the commutating integrals $K$ and $G$ generalizing $K_{\rm Y}$ and $G_{\rm RS}$, thus expressing in terms of these integrals the coefficients of the algebraic curve associated with the Lax representation: $$\begin{array}{l}
\displaystyle K= \left[\frac{1}{4}(M_1^2-M_2^2)+{\varepsilon}_0 (\alpha_1-\beta_2)+{\varepsilon}_1 [({{\boldsymbol \alpha}}{\times}{\mathbf{M}}){\bdot}\mathbf{e}_1-({{\boldsymbol \beta}}{\times}{\mathbf{M}}){\bdot}\mathbf{e}_2-{\varepsilon}_1 ({{\boldsymbol \alpha}}^2-{{\boldsymbol \beta}}^2)]     \right]^2\\[3mm]
\displaystyle \quad +\left[\frac{1}{2}M_1 M_2+{\varepsilon}_0 (\alpha_2+\beta_1)+{\varepsilon}_1 [({{\boldsymbol \alpha}}{\times}{\mathbf{M}}){\bdot}\mathbf{e}_2+({{\boldsymbol \beta}}{\times}{\mathbf{M}}){\bdot}\mathbf{e}_1-2{\varepsilon}_1  {{\boldsymbol \alpha}}{\bdot}{{\boldsymbol \beta}}]    \right]^2\\[3mm]
\displaystyle \quad +2\lambda \left[\frac{1}{4}(M_1^2+M_2^2)(M_3-2\lambda)+{\varepsilon}_0 (\alpha_3 M_1+\beta_3 M_2) \right] \\[3mm]
\displaystyle \quad -2 {\varepsilon}_1  \lambda \left[ \alpha_2 M_1^2-\beta_1 M_2^2-(\alpha_1-\beta_2)M_1 M_2+{\varepsilon}_1 ({{\boldsymbol \alpha}}^2+{{\boldsymbol \beta}}^2)-2 {\varepsilon}_1 ({{\boldsymbol \alpha}}{\times}{{\boldsymbol \beta}}){\bdot} \mathbf{M}\right], \\[3mm]
\displaystyle G= \frac{1}{4}\left[(\mathbf{M} {\bdot}{{\boldsymbol \alpha}})^2+ (\mathbf{M} {\bdot}{{\boldsymbol \beta}})^2\right]+\frac{1}{2}(M_3-2\lambda)({{\boldsymbol \alpha}}{\times}{{\boldsymbol \beta}}) {\bdot}\mathbf{M} \\[3mm]
\displaystyle \quad +{\varepsilon}_0 \left[{{\boldsymbol \beta}}^2\alpha_1 + {{\boldsymbol \alpha}}^2 \beta_2 -({{\boldsymbol \alpha}}{\bdot}{{\boldsymbol \beta}})(\alpha_2+\beta_1)\right]\\[3mm]
\displaystyle \quad -{\varepsilon}_1  \left[{{\boldsymbol \beta}}^2 ({{\boldsymbol \alpha}}{\times}\mathbf{M}){\bdot}\mathbf{e}_1+ {{\boldsymbol \alpha}}^2 ({{\boldsymbol \beta}}{\times}\mathbf{M}){\bdot}\mathbf{e}_2-({{\boldsymbol \alpha}}{\bdot}{{\boldsymbol \beta}})[({{\boldsymbol \alpha}}{\times}\mathbf{M}){\bdot}\mathbf{e}_2+({{\boldsymbol \beta}}{\times}\mathbf{M}){\bdot}\mathbf{e}_1 ]               \right]  \\[3mm]
\displaystyle \quad = \frac{1}{4}[(\mathbf{M} {\bdot}{{\boldsymbol \alpha}})^2+ (\mathbf{M} {\bdot}{{\boldsymbol \beta}})^2]+\frac{1}{2}(M_3-2\lambda)({{\boldsymbol \alpha}}{\times}{{\boldsymbol \beta}}) {\bdot}\mathbf{M} \\[3mm]
\displaystyle \quad +{\varepsilon}_0 ({{\boldsymbol \alpha}}{\times}{{\boldsymbol \beta}}){\bdot} \left[\mathbf{e}_2{\times}{{\boldsymbol \alpha}}+\mathbf{e}_1{\times}{{\boldsymbol \beta}}\right]-{\varepsilon}_1  \mathbf{M} {\bdot} \left[\mathbf{e}_1{\times}\bigl({{\boldsymbol \beta}}{\times}({{\boldsymbol \alpha}}{\times}{{\boldsymbol \beta}})\bigr)+ \mathbf{e}_2{\times}\bigl({{\boldsymbol \alpha}}{\times}({{\boldsymbol \alpha}}{\times}{{\boldsymbol \beta}})\bigr) \right].
\end{array}$$

Next, in the work [@RyabTMF] for the case $\lambda=0$ four critical subsystems of rank 2 are pointed out. For these subsystems, the constraints on the integral constants 