\qquad {{\boldsymbol \beta}}\ne 0 \qquad (\lambda={\rm const})$$ H.M.Yehia [@Yeh] found the integral $$\begin{array}{l}
K_{\rm Y} = (\omega _1^2  - \omega _2^2  + \alpha _1  - \beta _2 )^2 +
(2\omega _1 \omega _2  + \alpha _2  + \beta _1 )^2 \\[2mm]
\phantom{K = (}+ 2\lambda[(\omega_3-\lambda) ( \omega_1^2+ \omega_2
^ 2)
+ 2 \omega_1 \alpha_3 + 2 \omega_2 \beta_3 ].
\end{array}$$ Here the constant vector of the gyrostatic momentum is ${\boldsymbol \lambda} = \lambda \mathbf{e}_3$. For simplicity, the term “gyrostatic momentum” in this case is used for the scalar parameter $\lambda$. If in addition we suppose ${{\boldsymbol \beta}}\equiv 0$, then there exists the general first integral $$\begin{array}{l}
  L_{\rm Y} =  \alpha_1 \omega_1 + \alpha_2 \omega_2+ {\displaystyle \frac{1}{2}} \alpha_3 (\omega_3+\lambda)
\end{array}$$ similar to and the system becomes completely integrable. This case is known as the Kowaslevski–Yehia gyrostat.

In [@ReySem] a Lax representation of equations in the case was found that gave rise to the general first integral $$\begin{array}{l}
G_{\rm RS}= \left[\alpha_1 \omega_1+\alpha_2 \omega_2+\displaystyle \frac{1}{2}\alpha_3 (\omega_3+\lambda)\right]^2+ \left[\beta_1 \omega_1+\beta_2 \omega_2+\displaystyle \frac{1}{2}\beta_3 (\omega_3+\lambda)\right]^2 \\
\phantom{G_{\rm RS}} +(\omega_3-\lambda) \left[(\alpha_2 \beta_3-\alpha_3 \beta_2) \omega_1+(\alpha_3 \beta_1-\alpha_1 \beta_3) \omega_2+\displaystyle \frac{1}{2} (\alpha_1 \beta_2-\alpha_2 \beta_1) (\omega_3+\lambda)\right] \\
\phantom{G_{\rm RS}}  -{{\boldsymbol \beta}}^2 \alpha _1  - {{\boldsymbol \alpha}}^2 \beta_2 + {{\boldsymbol \alpha}}{\bdot} {{\boldsymbol \beta}}(\alpha_2+\beta_1).
