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Calculations of characteristics of stars with axial rotation in the frame of polytropic model are based on the solution of mechanical equilibrium equation – differential equation of second order in partial derivatives.  Different variants of approximate determinations of integration constants are based on traditional in the theory of stellar surface approximation, namely continuity of gravitational potential in the surface vicinity.  We proposed a new approach, in which we used simultaneously differential and integral forms of equilibrium equations.  This is a closed system and allows us to define in self-consistent way integration constants, the polytrope surface shape and distribution of matter over volume of a star.  With the examples of polytropes n=0 and n=1, we established the existence of two rotation modes (with small and large eccentricities).  It is proved that the polytrope surface is the surface of homogeneous rotational ellipsoid for the case n=0.  The polytrope characteristics with n=1 in different approximations were calculated as the functions of angular velocity.  For the first time it has been calculated the deviation of polytrope surface at fixed value of angular velocity from the surface of associated rotational ellipsoid.

Method of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n=0 and n=1