Three dynamical systems described by the systems of two differential equations with time and one spatial variables simulating vibrations of the double-walled carbon nanotubes conveying fluids are considered. The aim of this article is to develop the combination of Lie symmetry analysis and the dynamical system methods for dealing with symmetries, symmetry reductions and exact solutions to the carbon nanotube systems. At first, all of the geometric vector fields of the systems are presented with respect to the arbitrary parameters. Further the symmetry reductions and exact solutions to the systems are considered and investigated. On the base of such combination the exact power series solutions are obtained and their convergence proved.