Phase space of a dynamical system is the set of all admissible pairs (x, v) of coordinates and velocities for system of ODE equations. Mechanical system is called integrable if it has a function (first integral) that does not change along every trajectory.
Phase space of an integrable Hamiltonian system with two degrees of freedom (e.g. R^4) can be foliated on two-dimensional common level surfaces of the Hamiltonian and the first integral. Such fiber is usually the closure of the trajectories of the system.
Topological Fomenko--Zieschang invariants (finite graphs with numerical marks) of such foliations on 3-dimensional isoenergy surface Q^3 (submanifolds of fixed energy in the phase space) will be briefly discussed. Regular fiber is a 2-dimensional torus. They describe bifurcations of this foliation, i.e. the behaviour "near" a fiber with a critical trajectory.
These invariants were calculated for a family of analogs of famous Kovalevskaya integrable case in rigid body dynamics. Some of them are equivalent to invariants of other integrable systems. Recall that two systems have equivalent invariants if and only if their closures of trajectories have the same structure.