Method of critical subsystems as a way to calculate the types of critical points in integrable systems with three degrees of freedom

Consider an integrable Hamiltonian system with three degrees of freedom and its integral mapping defined by three functionally independent first integrals in involution. The critical set of this mapping is a union of the so-called critical subsystems, which are almost Hamiltonian systems with less than three degrees of freedom. Critical subsystems are described in two ways. First, they are defined as the sets of critical points lying on the zero level of some naturally arising general first integral. Second, the phase spaces of such critical subsystems are described by the pair of invariant relations. Using this integral and the invariant relations one can explicitly calculate the eigenvalues of the corresponding symplectic operator, thus obtaining the type of critical points belonging to the subsystem with respect to transversal cross-sections (the outer type). If the rank of some critical point is less than two, then it belongs to two or three critical subsystems and the corresponding outer types form the complete type of the point.
In this talk, we investigate the integrable Hamiltonian system with three degrees of freedom found by V. V. Sokolov and A. V. Tsiganov [1]. This system is known as the generalized two-field gyrostat. We find the pairs of the invariant relations describing invariant 4-dimensional manifolds bearing the critical subsystems which generalize the famous Appelrot classes of critical motions of the Kowalevski top [2].
For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial integrable system with three degrees of freedom.
The work is partially supported by RFBR, research project No. 14-01-00119.