Exact solutions of equations of rotationally symmetric motion of an ideal incompressible liquid

A partially invariant solution of the Euler equations is considered, where the vertical component of velocity is a function of the vertical coordinate and time, whereas the remaining components of velocity and pressure are independent of the polar angle in a cylindrical coordinate system. Using the classification of equations obtained by analysis of an overdetermined system, we consider two hyperbolic systems: the first one describes the motion of a cylindrical layer of an ideal incompressible liquid under a punch, and the second system allows obtaining solutions in a half-cylinder with singularities at the axis of symmetry. A class of new exact solutions is obtained, which describe vortex motion of an ideal incompressible liquid, including the motion with singularities (sources of vortices) located along the axis of symmetry.