  In this paper, we introduce a method of imposing asymmetric conditions on the
velocity vector with respect to independent variables and a method of moving
frame for solving the three dimensional Navier-Stokes equations. Seven families
of non-steady rotating asymmetric solutions with various parameters are
obtained. In particular, one family of solutions blow up at any point on a
moving plane with a line deleted, which may be used to study turbulence. Using
Fourier expansion and two families of our solutions, one can obtain
discontinuous solutions that may be useful in study of shock waves. Another
family of solutions are partially cylindrical invariant, contain two parameter
functions of $t$ and structurally depend on two arbitrary polynomials, which
may be used to describe incompressible fluid in a nozzle. Most of our solutions
are globally analytic with respect to spacial variables.
