The group-theoretic analysis of the Navier–Stokes equations in rotationally symmetric case and some new exact solutions

In this paper the group-theoretic properties (in the sence of S. Lie) of the Navier–Stokes equations in $R^3$ in rotationally symmetric case are studied. The main algebra of the equations (i.e., the maximal algebra of vector fields on the manifold of independent and dependent variables, whose flows leave the equations invariant) is computed. This algebra turns out to be infinite dimensional. The so-called optimal systems of subalgebras of dimension 1 and 2 of the main algebra are found and for the subalgebras of these systems corresponding reduced equations for the invariant solutions of the original equations are written. From this reduced equations some new exact solutions of the Navier–Stokes equations are obtained.