  In the first part of this article, we obtain a linear system whose the
solution solves the time-independent incompressible Navier-Stokes system for
the special case in which the external forces vector is a gradient. In a second
step we develop approximate solutions, also for the time independent
incompressible Navier-Stokes system, through the generalized method of lines.
We recall that for such a method, the domain of the partial differential
equation in question is discretized in lines and the concerning solution is
written on these lines as functions of the boundary conditions and boundary
shape. Finally, we emphasize these last main results are established through
applications of the Banach fixed point theorem.
