<p>Our work investigates the dynamical characteristics of a microtubule model, which arises from its intrinsic dipolar nature. The model assumes a single angular degree of freedom per dimer, describing the conformational displacements of constituent dimers in the radial direction. The integrability of this model is analyzed using the Painlevé singularity structure analysis, and our findings confirm that the model is integrable in the Painlevé sense. Additionally, the model is examined using Lie group analysis. The Lie point symmetries are determined under the invariance criteria of Lie groups, and the symmetry group corresponding to each generator is reported. Furthermore, the one-dimensional optimal system of sub algebras is constructed using the adjoint technique. The similarity reductions associated with nontrivial vector fields in the optimal system are then computed. By applying the similarity reduction method, the governing partial differential equation (PDE) is transformed into an ordinary differential equation (ODE) that includes a damping term dependent on the viscosity coefficient, the electric dipole moment, and the intrinsic dipolar electric field. This ODE is then reduced to a planar dynamical system. A bifurcation analysis is conducted in the case of the presence of the damping and in the case of its absence. In the presence of damping, Bendixson’s criterion is used to demonstrate the non-existence of closed phase orbits, which is further confirmed by the phase portraits. To explore possible solutions, the auxiliary function method is employed, yielding only kink and anti-kink solutions. Additionally, a power series method is used to obtain a series solution. These solutions are presented graphically, and the effects of the physical parameters on them are illustrated. On the other hand, in the absence of the damping term, additional different types of solutions are derived.</p>