  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.
