Abstract
This study explores a fourth-order nonlinear symmetric solution to ionic waves in living microtubules. Lie group analysis and the new extended direct algebraic approach are used to build solitary wave solutions. We reduce the partial differential equation based on symmetry into ordinary differential equations. Solved equations yield novel single wave patterns. For physical explanation, specific solitary wave structures are visually shown. The solutions include the anti-kink, kink shape, the absolute value of the spike shape, singular kink wave shape, and dark-singular soliton solution. Utilizing the multiplier technique, we establish the equation's local conservation laws. Some solution sketches depict the model's vision.