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The perturbations of the laminar shear-thinning viscoelastic pipe flow under Finitely Extensible Nonlinear Elastic model with Peterlin approximation (FENE-P) are shown to exhibit leading-order power-law behaviours, and the expected odd-even parities with respect to the radial coordinate that depend on the azimuthal wave\-number, $n$. The analysis helps regularizing the governing system of equations at the centreline, and allows for a complete stability analysis of three-dimensional perturbations for a general integer value of $n$, which has hitherto remained a challenge for FENE-P models. It is shown here that the symmetry and analytic behaviours of the velocity and pressure fields of the Newtonian counterpart are both preserved in this flow, and the reason is elucidated. For $|n|=1$, the perturbations to the correlations between the axial component and the radial or azimuthal components of the end-to-end polymer vector exhibit behaviour similar to that of the velocity perturbations close to the centreline, and are traced to the uniformity of axial traction with respect to the azimuthal direction. For all values of $n$, the fluctuation to the end-to-end length of the polymer chain vanishes at the centreline. The ansatzes for the perturbations to the components of conformation tensor arrived here, using heuristics, are later proved in two separate ways: Frobenius method, and a method that utilizes observations from Fourier analysis. We also reveal the existence of natural modes of polymers with trivial velocity perturbations in the limit of vanishing polymer viscosity. The complex frequency spectrum of these modes is continuous with radial stratification. Finally, the ansatzes are implemented using a Petrov--Galerkin spectral scheme.