  The incompressible Navier-Stokes equations are considered. We find that these
equations have symplectic symmetry structures. Two linearly independent
symplectic symmetries form moving frame. The velocity vector possesses
symplectic representation in a moving frame. The symplectic representation of
two-dimensional Navier-Stokes equations holds radial symmetry persistence. On
the other hand, we establish some results of radial symmetry either persistence
or breaking for the symplectic representations of three-dimensional
Navier-Stokes equations. Thanks radial symmetry persistence, we construct
infinite non-trivial solutions of static Euler equations with given boundary
condition.
