An example from the field of rigid body dynamics, possessing a natural physical justification, is presented. The behaviour of the solutions of the equations of motion in the real domain, whatever the initial data, is regular; nevertheless, depending on the values of a certain control parameter, the solution of the system may branch in the complex time plane, and the system will have multi-valued first integrals. A denumerable sequence of single-valued polynomial integrals of arbitrarily high even degree is found (unlike Kovalevskaya's case, in which the degree of the first integral of the Euler–Poisson equations is four). As an extension, a system from non-holonomic mechanics is considered.

In the frame of this work the Kovalevski exponents (KE) have been found for various problems arising in rigid body dynamics and vortex dynamics. The relations with the parameters of the system are shown at which KE are integers. As it is shown earlier the power of quasihomogeneous integral in quasihomogeneous systems of differential equations is equal to one of IKs. That let us to find the power of an additional integral for the dynamical systems studied in this work and then find it in the explicit form for one of the classic problems of rigid body dynamics. This integral has an arbitrary even power relative to phase variables and the highest complexity among all the first integrals found before in classic dynamics (in Kovalevski case the power of the missing first integral is equal to four). The example of a many-valued integral in one of the dynamic systems is given.