Lagrangian invariants in the equilibrium conditions of an ideal rotating fluid

It is shown that the velocity distribution and equilibrium conditions of the self-gravitating configurations from an ideal fluid are closely related with the presence of freezing-in vector fields determined by means of scalar potentials, whose total derivative is equal to zero. Ignorance of integrals of motion connected with the existence of freezing-in fields did not allow one to obtain the complete equilibrium conditions by variational method and, especially, led to the divergence in Lyapunov's and Riemann's results. This defect is eliminated in the paper and the equilibrium conditions and velocity distribution are considered by a single approach in the models of: (1) homogeneous fluid; (2) compressible fluid; (3) fluid with a magnetic field; (4) ideal fluid in the PNA of GR.