\omega_2=0.$$ Since $\omega_2=0$ yields $\dot \omega_1=0$, to obtain the system of invariant relations we replace the first condition in with $\dot \omega_2=0$. Thus, the equations of the corresponding manifold in ${\mathcal{P}}^5$ are $$\label{eq3_7}
  \omega_1 \omega_3+\alpha_2 =0, \qquad \omega_2=0.$$

In the following sections we point out the equations of three invariant manifolds generalizing the Appelrot classes for the Hamilton function . Two of them exist only for the top with $\lambda=0$ and correspond to the bifurcation surfaces found in [@RyabTMF], the third one exists for arbitrary values of $\lambda$ and generalizes the cases shown in the works [@KhRCD09; @KhJPA; @KhHMJ].

The extension of the 1st Appelrot class {#sec3}
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It is known that the 1st Appelrot class is not stable with respect to the gyrostatic momentum. Indeed, already for the case of a single field (the gravity field) the corresponding invariant manifold disappears for the Kowalevski–Yehia gyrostat with $\lambda \ne 0$ (see [@Ryab1]). To illustrate this fact, put ${{\boldsymbol \beta}}=0, {\varepsilon}_1 =0$ and consider the image of the first three Appelrot classes in the space of the integral constants ${\mathbb{R}}^3(\ell,h,k)$. It is a part of the bifurcation diagram of the corresponding integral map. The cross-section of this set by the plane 