  In the present report, by using the Stokes-Helmholtz decomposition theorem
the 3-dimensional Navier-Stokes equation (NSE) is uncoupled and transformed
into a scalar equation for the velocity potential when the flow field is
toroidal. The dynamics of the velocity potential is independent of the vector
potential. The reduction and invariant solutions to the equation are analyzed
by the Lie symmetry method subsequently. The Lie subalgebras for the equation
are discussed and the corresponding invariant solutions are also presented.
