Integrable models of rotationally symmetric motions of an ideal incompressible fluid

A study has been carried out to find the exact solutions to the equations of the rotationally symmetric motion of an ideal incompressible fluid. Equations possess a large group of admissible transformations, which yield most of the exact solutions to them. Since the conversion to the new unknown functions in a system is not a local transformation, the Lie algebras admitted by the systems are not necessarily isomorphic. Direct computations suggest that the Lie algebra corresponding to the largest group admitted by the system is generated by the mathematical operators. A characteristic feature of these solutions is the occurrence of a singularity of function in a finite time. As the new unknown functions are introduced and the system is reduced to a form, the first two equations form an inhomogeneous Cauchy-Riemann system for the functions, while the last two equations form a system of hyperbolic equations for the functions.