The paper discusses relationship between regular behavior of Hamilton's systems and the existence a sufficient number of fields of symmetry. Some properties of quite regular schemes and their relationship with various characteristics of stochastic behavior are studied.

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.

It has been discovered a countable number of dynamic systems with an equal countable set of the first integrals and invariant measure. The found systems are a generalization of so-called Manakov's systems on $SO(n)$ algebra and the integrable Chaplygin's problem about ball rolling.

There is shown in this work that in the case when the configuration space of a dynamic system is a two-dimensional sphere, any field of symmetries of the second order is Hamiltonian as to its pulses. There is described a large class of fields of symmetries of the geodesic flow on the two-phase sphere.

There has been studied problem of self-similarity of periodical trajectories in Hamilton's systems in the infinite sequence of bifurcations of duplication of the period using as an example the problem of oscillations of satellite on the elliptic orbit relative to the proper center of mass. The universal scale regularities of transformation of periodical movements of the system within the limits of the chaotic movement have been discovered using methods of continuation of solution according to the parameter, the theory of stability by Ljapunov and Floquet, the theory of branching and the methods of scaling, as well. There has been suggested a numeric algorithm of building of the scaling functions of the trajectories (STF) in Hamilton's systems and determination on their basis of universal scale factors of transition to chaos. It has been shown that the STF of Hamilton's systems have a range of qualitative and quantitative dissimilarities from the known dissipative analog.

The paper studies the behavior of a passive addition in the field of speed of three blowing-up point vortexes and also in interaction of three distributed vortex spots of a similar vorticity in two-dimensional incompressible endless medium are. The vortex spots are considered in the frame of a moment model of the second order where they are described by the elliptic Kirchhoff vortexes.

There is discussed one billiard problem describing an interaction of two mass points united by an non-extensible weightless thread the center of mass of which moves along the circular Kepler orbit. There have been built phase portraits of Poincare's maps and found the fields of the regular behavior of the system.

The question of the integrability of the mixmaster model of the Universe, presented as a dynamical system with finite degrees of freedom, is investigated in the present paper.