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For the first time in 250 years since statement of the Solar planetary system formation problem accompanied by formulation and rejection of numerous hypotheses and theories, correct theory is developed, which provides insight into most of laws of the Solar system structure and motion. Moreover, this theory is suitable for research of the exoplanetary systems of other stars, for which it allows theoretical identification of the complete system structure. A number of issues of astrophysics related to stellar and galactic systems has been successfully resolved based on analytical solution of hydrodynamic equations. In particular, the paradox of incompatibility of mass and angular momentum of bodies in the Solar System has been overcome; the Planetary Distances Law, which is a theoretical analogue of the Titius-Bode empirical law, has been received; real charts of galactic rotation curves have been plotted, etc. Finally, we show that the vortex theory provides in some of its consequences new approaches to the physical essence elucidation of the paradoxical and ambiguous objects and phenomena such as "dark energy", "dark matter", the Big Bang and the Universe expansion.