An Integrable Model of Nonstationary Rotationally Symmetrical Motion of Ideal Incompressible Liquid

We consider a partially invariant solution of the Eulerequations with respect to a six-parameter Lie group admitted by thissystem where the vertical component of velocity is a function of thevertical coordinate and time only while two other components andpressure do not depend on the polar angle in a cylindrical coordinatesystem. The analysis of the corresponding overdetermined system leads totheir special (but nontrivial) dependence of the polar radius. Afterthis, the nonlinear factor-system for invariants of the group is reducedto a system of ordinary differential equations by introduction ofLagrangian coordinates. As a result, we obtain a wide class of new exactsolutions which describes vortex motions of an ideal incompressibleliquid including motions with singularities.