given by the equations $$\label{eq2_12}
\Phi_{\mathcal{L}}=0, \qquad \rmd \Phi_{\mathcal{L}}=0$$ and the induced dynamical system on it is almost everywhere Hamiltonian with two degrees of freedom. The global Hamiltonian structure may have singularities at the points where either $\mathcal{J}_{\mathcal{L}}$ fails to be smooth or the 2-form induced on $\mathcal{J}_{\mathcal{L}}$ by the symplectic structure degenerates. Critical subsystems of rank 1 (with one degree of freedom) can be obtained in a similar way starting with two equations of the type .

In analytical case (particularly, in the case when all functions and equations have polynomial structure) critical subsystems are defined uniquely if we consider all irreducible equations of the type . Such systems generate a stratification of the phase space and determine bifurcations that occur to the Liouville tori. Invariant relations describing a critical subsystem make it possible to explicitly calculate the types of critical points for all ranks.

Critical subsystems were found for the Kowalevski top in a double field in [@Bogo; @Kh32; @KhRCD] and for the Kowalevski gyrostat in a double field in [@KhJPA; @KhHMJ]. In both cases, one of these subsystems has one degree of freedom and is formed completely by the pendulum type motions. For the top, there exist 