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       Integrable and stochastic motions in the N---body problem}\\
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        {\sl L.L.Sokolov \\
        Astronomical Institute St.Petersburg University,\\
        Stary Peterhof, 198904 St.~Petersburg, Russia.\\
        E-mail: lsok@aispbu.spb.su}
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The well--known results of Poincare and Bruns set limits on the global
integrability of the N--body problem. But the regions exist with
simple structure of trajectories. Using the Picard's iterations technique,
one can find large areas, where the N--body problem is integrable in
classical sense. These bodies have large relative velocities and the
close approaches are absent. The motions near collisions are more
complicated and call for a specific technique. Let the planets move
in frameworks of the KAM--theory, zero-mass body travels near orbits
with collisions. Convenient approximation is point--like action sphere
method. The contraction of action sphere to the point formally leads
to indeterministic (stochastic) description of motion by means of tree
like Markov process. The result of tree properties investigation is
the evidence for presence of trajectory passing near every Keplerian
elliptic orbit intersecting any noncircular planetary orbit. In the
small neighbourhood of the stochastic approximation one can find the
exact solution of the equations of motion; the method of such
solution construction is elaborated. For example, the restricted planar
circular three-body problem has three resonance collision orbits,
the ratios of periods are equal to 1/1, 1/2, 2/1 . Arbitrary infinite sequence
containing three symbols 1/1, 1/2, 2/1 (successive symbols are different)
corresponds to point--like action sphere method trajectory. Using
Jacobi's integral one can prove the existence of exact solution of
the restricted three-body problem, which is close to such arbitrary
sequence for all time.


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